AN   ADVANCED   COURSE   OF   INSTRUCTION 
IN 

CHEMICAL    PRINCIPLES 


THE    MACMILLAN   COMPANY 

NEW  YORK    •    BOSTON   •    CHICAGO   •    DALLAS 
ATLANTA   •    SAN  FRANCISCO 

MACMILLAN  &  CO..  LIMITED 

LONDON   •    BOMBAY   •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


AN  ADVANCED  COURSE  OF  INSTRUCTION 

IN 

CHEMICAL    PRINCIPLES 


, 

£ 

ARTHUR  A.  NOYES 

DIRECTOR   OF   THE   GATES   CHEMICAL  LABORATORY 
(CALIFORNIA  INSTITUTE  OF   TECHNOLOGY 

AND 

MILES  S.  SHERRILL 

ASSOCIATE  PROFESSOR   OF   THEORETICAL   CHEMISTRY 
MASSACHUSETTS   INSTITUTE   OF  TECHNOLOGY 


NEW  YORK 

THE   MACMILLAN  COMPANY 
1922 


PRINTED  IN  THE   UNITED  STATES   OF   AMERICA 


PRELIMINARY  EDITIONS 
COPYRIGHT,  1017  AND  iQ2o, 
BY  ARTHUR  A.  NOYES 


COMPLETE  REVISION 

COPYRIGHT,  1022, 

BY  ARTHUR  A.  NOYES  AND  MILES  S.  SHERRILL 

Set  up  and  electrotyped.     Published  May,  1922 


Press  of  J.  J.  Little  &  Ives  Co. 
New  York 


PREFACE 


IN  this  book  are  presented  the  results  of  the  authors'  many  years'  ex- 
perience with  their  own  classes  in  the  development  of  a  thorough  course  of 
instruction  in  the  laws  and  theories  of  chemistry  from  quantitative  stand- 
points. The  course  is  intended  for  junior,  senior,  or  graduate  students  of 
physical  chemistry  in  colleges,  scientific  schools,  and  universities,  who  have 
completed  the  usual  freshman  and  sophomore  courses  in  chemistry,  physics, 
and  mathematics. 

The  purpose  of  the  course  is  to  give  such  students  an  intimate  knowledge 
of  fundamental  chemical  principles  and  a  training  in  logical,  scientific 
thinking,  such  as  will  enable  them  to  attack  effectively  the  practical  prob- 
lems arising  in  their  subsequent  educational  or  professional  work  in  any  of 
the  branches  of  chemistry  or  related  sciences.  Descriptive  text-books  of 
physical  chemistry  afford  a  general  survey  of  chemical  laws  and  theories 
which  may  suffice  for  the  purposes  of  students  who  are  not  preparing  for 
a  professional  career  on  the  educational,  research,  or  industrial  sides  of 
chemistry,  but  they  do  not  give  that  intensive  training  which  is  essential 
for  pursuing  successfully  more  specialized  courses  of  scientific  study  or  for 
applying  chemical  principles  to  industrial  problems.  No  one  regards  a 
course  of  lectures  on  the  principles  of  mathematics  as  a  suitable  method  of 
giving  beginners  the  ability  to  handle  that  science,  and  with  scarcely  more 
reason  can  descriptive  courses  on  physical  chemistry  be  expected  to  afford 
a  working  knowledge  of  chemical  principles  in  their  quantitative  aspects. 
Only  by  constantly  applying  the  principles  to  concrete  problems  will  the 
student  acquire  such  a  knowledge  and  the  power  to  use  it  in  new  cases. 

Accordingly,  the  course  of  instruction  is  so  planned  as  to  make  the 
student  think  about  the  significance  of  the  principles  presented  and  work 
out  for  himself  the  method  of  treatment  of  special  cases  upon  the  basis 
of  those  principles.  To  this  end,  the  text  is  interspersed  with  problems, 
which  prevent  the  student  from  memorizing  the  principles  or  complacently 
believing  that  a  formal  knowledge  constitutes  a  real  understanding  of  them. 
These  problems  are  for  the  most  part  not  of  the  usual  type,  involving  merely 
substitution  in  formulas  and  mathematical  operations,  but  are  such  as 
require  clear  logical  thinking  in  the  application  of  the  principles  to  the 
cases  under  consideration.  They  are  not  merely  supplementary  or  inci- 
dental to  the  text,  but  they  are  the  feature  about  which  the  whole  presen- 


yi  PREFACE 

tation  centers.  The  aim  striven  for  has  been  to  make  each  problem  serve 
a  definite  purpose,  and  to  have  it  involve  independent  thought,  yet  in  such 
measure  as  shall  not  be  beyond  the  mental  capacity  of  college  students  who 
have  completed  good  general  courses  in  mathematics,  physics,  and  chemistry. 
The  problems  have  been  gradually  developed  as  a  result  of  many  years' 
trial  of  the  plan  with  large  classes  of  such  students.  While  the  authors 
realize  that  the  presentation  is  in  its  details  still  far  from  perfect,  they 
have  decided  to  defer  publication  of  it  no  longer,  in  the  hope  that  this 
educational  method,  novel  in  this  subject,  may  be  further  developed  through 
the  experience  of  other  teachers,  from  whom  suggestions  will  be  gratefully 
received. 

In  order  to  attain  the  purpose  in  view  and  yet  keep  the  book  within  the 
limits  of  an  undergraduate  course,  the  subjects  treated  have  been  carefully 
selected  from  the  point  of  view  of  their  practical  importance  to  the  chemist; 
many  topics  being  omitted  that  are  commonly  included  in  descriptive 
courses  on  physical  chemistry  or  in  complete  treatises  on  the  subject.  The 
book  consists  mainly,  as  the  table  of  contents  shows,  in  a  development  of 
the  atomic,  kinetic,  and  ionic  theories  through  a  consideration  of  the  physical 
properties  directly  related  to  them,  and  in  a  treatment,  with  the  aid  of  these 
theories,  from  mass-action,  phase,  and  thermodynamic  view-points,  of  the 
principles  relating  to  the  rate  and  equilibrium  of  chemical  reactions.  The 
newer  theories  of  atomic  structure  and  of  radiation  have  been  reluctantly 
omitted;  because,  in  spite  of  their  transcendent  interest  and  their  im- 
portance to  the  future  of  chemistry,  in  their  present  stage  of  development 
they  are  appropriately  treated  in  an  advanced  course  of  the  research  type, 
rather  than  in  a  fundamental  undergraduate  course  of  a  systematic  character. 
Lack  of  tune  has  led  also  to  the  omission  of  those  more  specialized  parts 
of  the  subject  that  treat  of  the  physical  properties  of  substances  in  the 
various  states  of  aggregation,  such  as  the  metallic,  crystalline,  liquid,  and 
dispersed  states;  for,  important  as  are  some  of  the  generalizations  already 
arrived  at  in  these  fields,  their  inclusion  in  this  course  would  have  involved 
sacrifice  of  an  adequate  and  intensive  treatment  of  the  more  fundamental 
principles.  A  systematic  notation  (summarized  on  page  297)  has  been 
employed  throughout  the  book. 

The  course  of  instruction,  as  carried  out  by  the  authors,  consists  of 
class-room  exercises,  mainly  of  the  recitation  type,  given  to  sections  of 
twenty  to  thirty  students.  At  each  exercise  a  number  of  problems  are 
assigned  which  are  to  be  solved  and  handed  in  at  the  beginning  of  the  next 
exercise;  the  students  being  advised  to  use  slide-rules  so  as  to  shorten  the 
arithmetical  work.  The  instructor  then  solves  these  problems  on  the 
blackboard,  questioning  members  of  the  class  as  to  how  he  shall  proceed. 
He  emphasizes  the  principles  involved  and  the  best  way  of  looking  at  the 
problems  under  consideration*.  In  the  assignments  it  is  customary  not  to 


PREFACE  vii 

give  out  all  at  once  the  whole  group  of  problems  following  a  section  of  the 
text,  but  to  assign  only  the  first  one  or  two  of  such  a  group,  together  with 
the  last  problems  of  the  preceding  group.  This  enables  the  instructor  to 
discuss  the  principles  relating  to  a  new  group  of  problems  after  the  students 
have  given  some  thought  to  them,  but  before  they  have  passed  to  another 
topic;  thereby  insuring  a  better  understanding  of  the  general  view-point 
from  which  it  is  best  to  attack  the  problems  and  removing  difficulties  which 
individual  students  may  experience.  Frequent  written  tests  are  given,  con- 
sisting of  review  problems  different  from  those  already  solved  by  the  class, 
but  involving  the  same  principles. 

In  order  that  the  student  may  better  appreciate  the  principles  he  is 
studying,  it  is  desirable  that  the  class  work  be  accompanied  by  a  brief 
laboratory  course,  or  if  that  is  not  practicable  by  lecture  experiments,  whose 
primary  purpose  should  be  to  give  concrete  illustrations  of  the  nature  of  the 
basic  phenomena  under  consideration  (such  as  vapor-pressure,  ion-migration, 
reaction-rate);  for  the  experimental  methods  by  which  the  properties  are 
determined  are  not  described  in  this  book,  except  in  so  far  as  these  are  es- 
sential to  an  understanding  of  the  phenomena.  Moreover,  no  attempt 
has  been  made  to  present  the  historical  and  research  aspects  of  the  topics, 
because  of  the  impossibility  of  doing  this  at  all  adequately  in  a  brief  sys- 
tematic treatment  of  the  well-established  principles  of  the  subject.  Many 
teachers  will  doubtless  desire  to  supplement  this  systematic  presentation 
by  mentioning  these  historical  and  research  aspects,  and  by  suggesting  the 
collateral  reading  of  reviews  and  original  articles  that  have  appeared  in 
chemical  periodicals,  or  of  text-books  of  physical  chemistry  in  which  these 
aspects  are  more  fully  discussed. 

To  cover  satisfactorily  the  whole  subject  as  here  presented  requires  from 
120  to  150  exercises;  but,  in  order  to  make  it  readily  adaptable  to  a  one-year 
go-hour  course,  certain  articles  and  problems  which  are  less  important  or 
more  difficult  are  indicated  by  asterisks.  It  is  also  suggested  that,  when  the 
subject  must  be  completed  in  a  one-year  course,  no  attempt  be  made  to 
include  the  thermodynamic  treatment  presented  in  Chapters  X-XIII, 
but  that,  after  taking  up  the  chapter  on  thermochemistry,  the  course 
be  completed  by  considering  (without  reference  to  its  thermodynamic 
derivation)  the  applications  of  the  important  van't  Hoff  equation,  as  pre- 
sented in  Arts.  167-169.  When  the  course  extends  through  a  year  and  a 
half,  the  second  year  work  will  naturally  consist  of  Chapters  IX-XIII,  these 
constituting  a  systematic  course  on  thermodynamic  chemistry,  which  in- 
deed may  be  pursued  as  a  graduate  subject  by  those  who  have  not  studied 
the  earlier  chapters  of  the  book. 

The  authors  desire  to  express  their  indebtedness  to  many  of  their  col- 
leagues for  valuable  suggestions,  and  especially  to  Professors  S.  J.  Bates  and 
E.  B.  Millard,  and  to  Mr.  Roger  Williams,  who  have  furnished  detailed  lists 


viii  PREFACE 

of  corrections  and  general  criticisms.  The  authors  desire  also  to  thank 
Drs.  G.  N.  Lewis  and  Merle  Randall  for  the  privilege  of  including,  in  ad- 
vance of  its  publication  in  their  book  on  Thermodynamics  and  the  Free 
Energy  of  Chemical  Substances,  a  table  showing  the  values  of  the  electrode- 
potent^als  recently  derived  by  them. 
Pasadena  and  Cambridge,  March,  IQ22. 


CONTENTS 


PART  I 

THE  ATOMIC,  MOLECULAR,  AND  IONIC  THEORIES  AND 

THE  PROPERTIES  OF  SUBSTANCES  DIRECTLY 

RELATED  TO  THESE  THEORIES 


CHAPTER  I:    THE  COMPOSITION  OF  SUBSTANCES 
AND  THE  ATOMIC  THEORY 

PAGE 

1.  The  Field  of  Chemistry.    Its  General  Principles  the  Subject  of 

this  Course     .     .     .     .     ...    >.,,  , 3 

2.  Pure  Substances  and  Mixtures,  Elementary  and  Compound  Sub- 

stances, the  Elements,  and  the  Law  of  Definite  Proportions     .  3 

3.  The  Law  of  Combining  Weights 4 

4.  The  Atomic  Theory 5 

5.  Determination  of  Combining  and  Atomic  Weights       ....  6 

6.  Values  of  the  Atomic  Weights 8 

7.  Chemical  Formulas,  Formula- Weights,  and  Equivalent- Weights  8 

CHAPTER  II:    THE  MOLAL  PROPERTIES  OF  GASES 
AND  THE  MOLECULAR  AND  KINETIC  THEORIES 

I.  THE  VOLUME  OF  GASES  IN  RELATION  TO  PRESSURE,  TEMPERATURE, 
AND  MOLAL  COMPOSITION 

8.  The  Volume  of  Perfect   Gases   in   Relation   to   Pressure  and 

Temperature 10 

9.  Law  of  Combining  Volumes  and  the  Principle  of  Avogadro    .     .       12 

10.  Empirical  Definition  of  Molecular  Weight  and  of  Mol       ...  12 

11.  General  Expression  of  the  Laws  of  Perfect  Gases 13 

12.  Dalton's  Law  of  Partial  Pressures 14 

13.  Determination  of  Molecular  Weights 16 

14.  Derivation  of  the  Atomic  Weights  of  Elements  and  of  the  Mo- 

lecular Composition  of  Compounds 17 

15.  Deviations  from  the  Perfect-Gas  Laws  at  Moderate  Pressures     .       19 

16.  Pressure-Volume  Relations  of  Gases  at  High  Pressures     ...       19 


x  CONTENTS 

H.    THE   KINETIC  THEORY 

PAGE 

17.  The  Fundamental  Kinetic  Hypotheses 22 

18.  The  Kinetic  Equation  for  Perfect  Gases 22 

19.  The  Kinetic  Energy  of  the  Molecules  and  the  Avogadro  Number  24 

20.  The  Kinetic  Equation  for  Imperfect  Gases  . 25 

21.  The  Characteristics  of  the  Molecules 29 

22.  Distribution  of  the  Velocities  and  Kinetic  Energies  of  the  Mole- 

cules of  a  Gas      ....    V    .     . 31 

HI.    THE   ENERGY  RELATIONS   OF   GASES 

23.  Energy  in  General  and  the  Law  of  its  Conservation     ....       35 

24.  Work  Attending  Volume  Changes  in  General    .     .     .     .     .      .       37 

25.  The  Energy-Content  of  Systems  in  General  in  Relation  to  Tem- 

perature and  Heat- Capacity     .     .     .     , 39 

26.  The  Energy-Content  and  Heat-Capacity  of  Perfect  Gases      .      .       40 

27.  The  Heat- Capacity  of  Perfect  Gases  in  Relation  to  their  Mo- 

lecular Composition .  '  .     .     .     .     .       41 

28.  The  Energy-Content  and  Heat-Capacity  of  Perfect  Gases  in  Re- 

lation to  the  Kinetic  Theory 42 

29.  The  Energy-Content  of  Imperfect  Gases  in  Relation  to  Volume 

and  Pressure :     .  44 

30.  The  Energy-Content  of  Imperfect  Gases  at  Constant  Temperature 

in  Relation  to  the  Kinetic  Theory 45 

CHAPTER  III:   THE  MOLAL  PROPERTIES  OF  SOLUTIONS 
AND  THE  MOLECULAR  THEORY 

I.    VAPOR-PRESSURE  AND   BOILING-POINT   IN   GENERAL 

31.  Vapor-Pressure „ 47 

32.  Relation  of  Boiling-Point  to  Vapor-Pressure 48 

33.  Change  of  Vapor-Pressure  with  Temperature.    The  Clapeyron 

Equation 48 

34.  Vaporization  in  Relation  to  the  Kinetic  Theory 51 

II.     SOLUTIONS   IN   GENERAL 

35.  The  Nature  and  Composition  of  Solutions 55 

m.    VAPOR-PRESSURE  AND  BOILING-POINT  OF  PERI.ZCT  SOLUTIONS 
WITH  ONE  VOLATILE   COMPONENT 

36.  Raoult's  Law  of  Vapor-Pressure  Lowering 57 

37.  Relation  of  Boiling-Point  Raising  to  Vapor-Pressure  Lowering 

and  Molal  Composition 59 

38.  Determination  of  Molecular  Weights 62 

39.  Partial  Vapor-Pressure  of  Volatile  Solutes.    Henry's  Law      .      .       63 


CONTENTS  xi 

IV.      VAPOR-PRESSURE    AND    BOILING-POINT    OF    CONCENTRATED    SOLUTIONS 
WITH  TWO   VOLATILE   COMPONENTS 

PAGE 

40.  The  Vapor-Pressure  of  Concentrated  Solutions 67 

41.  Vapor-Pressure  and  Boiling-Point  of  Concentrated  Perfect  Solu- 

tions in  Relation  to  their  Molal  Composition      .....       67 

42.  Vapor-Pressure  of  Concentrated  Solutions  in  General  in  Relation 

to  their  Composition 69 

43.  Boiling-Point  of  Concentrated  Solutions  in  General  in  Relation  to 

their  Composition 72 

V.    DISTRIBUTION  BETWEEN  PARTIALLY  MISCIBLE   SOLVENTS 

44.  Determination    of    Equilibrium-Conditions    by    the    Perpetual- 

Motion  Principle 76 

45.  Distribution  of  a  Solute  between  Two  Non-Miscible  Solvents      .       76 

46.  The  Lowering  by  Solutes  of  the  Solubility  of  One  Solvent  in 

Another  Solvent 78 

47.  The  Vapor-Pressure  Relations  of  Partially  Miscible  Liquids  .     .       79 

VI.    THE   FREEZING-POINT   OF   SOLUTIONS 

48.  Freezing-Point  and  Its  Relation  to  Vapor-Pressure  and  Molal 

Composition 80 

49.  Determination  of  Molecular  Weights       ........       83 

VII.    OSMOTIC  PRESSURE  OF  SOLUTIONS 

50.  Osmotic  Pressure 84 

51.  Relation  of  Osmotic  Pressure  to  Vapor-Pressure 85 

52.  Relation  between  Osmotic  Pressure  and  Molal  Composition .      .       87 

Vin.    REVIEW  OF  THE  PRINCIPLES  RELATING  TO  THE  MOLAL  PROPERTIES 

53.  Review  Problems 89 


CHAPTER  IV:    THE  ATOMIC  PROPERTIES 
OF  SOLID  SUBSTANCES 

I.    HEAT-CAPACITIES   OF   SOLID   SUBSTANCES 

54.  Properties  in  the  Solid  State         .  91 

55.  The  Heat-Capacity  of  Solid  Elementary  Substances    ....  91 

56.  The  Heat-Capacity  of  Solid  Compound  Substances     ....  93 

57.  Determination  of  Atomic  Weights 94 

H.    GENERALIZATIONS   RELATING  TO  ATOMIC  WEIGHTS 

58.  Methods  of  Atomic  Weight  Determination  and  the  Periodic  Law      95 


xii  CONTENTS 

CHAPTER  V:  THE  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 
AND  THE  IONIC  THEORY 

I.    THE  MOLAL  PROPERTIES   OF   SOLUTIONS 
OF  IONIZED   SUBSTANCES 

PAGE 

59.  Effects  of  Salts  on  the  Molal  Properties  of  Aqueous  Solutions  and 

their  Explanation  by  the  Ionic  Theory 96 

H.    ELECTROLYSIS  AND  FARADAY'S  LAW 

60.  Electrolytic  Conduction 98 

61.  Chemical  Changes  at  the  Electrodes 98 

62.  Faraday's  Law 99 

63.  The  Mechanism  of  Conduction  between  Electrodes  and  Solutions  100 

m.    ELECTRICAL  TRANSFERENCE 

64.  The  Phenomenon  of  Electrical  Transference 104 

65.  The  Law  of  Transference 105 

66.  Transference-Numbers       .    - 105 

67.  The  Mechanism  of  Conduction  in  Solutions 106 

68.  Transference  in  Relation  to  the  Mobility  of  the  Ions  ....  107 

69.  The  Moving-Boundary  Method  of  Determining  Transference      .  108 

70.  Change  of  Transference-Numbers  with  the  Concentration      .     .  109 

71.  The  Composition  of  Ions  Determined  by  Transference  Experi- 

ments  no 

IV.    ELECTRICAL  CONDUCTANCE 

72.  Conductance,  Specific  Conductance,  and  Equivalent  Conductance  112 

73.  Conductance  in  Relation  to  the  Mobility  of  the  Ion-Constituents  113 

74.  Change  of  Equivalent  Conductance  with  the  Concentration  .      .  114 

75.  Conductance  in  Relation  to  the  Concentration  and  Mobility  of 

the  Ions .     .     ...     .     .     .     .      .     u6 

76.  Conductance  of  the  Separate  Ion-Constituents      - 119 

77.  Determination  of  the   Concentration  of  Ions  and  of  Largely 

Ionized  Substances  by  Conductance  Measurements       .      .      .     121 

V.    THE  IONIZATION  OF  SUBSTANCES 

78.  lonization  of  Substances  Not  Largely  Ionized   ...     .     .     .     123 

79.  lonization  of  Largely  Ionized  Substances     .......     123 

VI.    APPLICATIONS  OF  THE  LAWS   OF  CONDUCTANCE 
AND  TRANSFERENCE 

80.  Review  Problems    .     . 127 


CONTENTS  xiii 

PART  II 

THE  RATE  AND  EQUILIBRIUM  OF  CHEMICAL  CHANGES 
FROM  MASS-ACTION  AND  PHASE  VIEW-POINTS 

CHAPTER  VI:    THE  RATE  OF  CHEMICAL   CHANGES 

I.    THE    EFFECT    OF   CONCENTRATION   ON   REACTIONS 
BETWEEN   SOLUTES    OR    GASEOUS   SUBSTANCES 

PAGE 

81.  Concept  of  Reaction-Rate 131 

82.  The  Mass-Action  Law  of  Reaction-Rate  between  Solutes       .     .  131 

83.  First-Order  Reactions 132 

84.  Second-Order  and  Third-Order  Reactions 133 

85.  The  Mechanisn^  of  Reactions  and  the  General  Mass-Action  Law 

of  Reaction-Rate 134 

86.  The  Mass-Action  Law  of  Reaction-Rate  between  Gaseous  Sub- 

stances       135 

87.  Simultaneous  Reactions 136 

II.    THE  EFFECT  OF  CONCENTRATION  AND  SURFACE  ON  REACTIONS 
BETWEEN  SOLUTES  AND   SOLID   SUBSTANCES 

88.  Rate  of  Reaction  between  Solutes  and  Solid  Substances    .     .     .     139 

89.  Solid  Substances  Dissolving  in  their  Own  Solutions     ....     139 

in.    THE  EFFECT   OF  CATALYSTS 

90.  Catalysis .....     .     .     .     .  140 

91.  Carriers .  140 

92.  Contact  Agents        .      .      .      .     .     .     .     .    (.     .     .     .     .     .  141 

93.  Hydrogen-Ion  and  Hydroxide-Ion  as  Catalysts       .     .     .     .     .  142 

94.  Enzymes 143 

95.  Water  as  a  Catalyst      ..............  143 

IV.    THE   EFFECT   OF  TEMPERATURE 

96.  Principles  Relating  to  the  Effect  of  Temperature 144 


CHAPTER  VII:    THE  EQUILIBRIUM  OF  CHEMICAL  CHANGES 
AT  CONSTANT  TEMPERATURE 

I.    THE  LAW   OF  MASS-ACTION 

97.  The  Equilibrium  of  Chemical  Reactions 145 

98.  The  Mass-Action  Law  of  Chemical  Equilibrium 146 


xiv  CONTENTS 

H.    THE   MASS-ACTION   LAW   FOR   REACTIONS   BETWEEN   GASES 

PAGE 

99.   The  Mass-Action  Law  in  Terms  of  Partial  Pressures       .     .      .  148 

100.  Gaseous  Dissociation 148 

101.  Metathetical  Gas  Reactions 150 

m.    THE   MASS-ACTION   LAW   FOR  REACTIONS   BETWEEN   SOLUTES 

102.  lonization  of  Slightly  Ionized  Univalent  Acids  and  Bases     .      .  152 

103.  lonization  of  Largely  Ionized  Substances    . 153 

104.  The  lonization  of  Water .      .      .  154 

105.  The  Hydrolysis  of  Salts 154 

1 06.  lonization  of  Dibasic  Acids  and  their  Acid  Salts 157 

107.  Displacement  of  One  Acid  or  Base  from  its  Salt  by  Another      .  159 

108.  Neutralization-Indicators       .     .     ...     . 161 

IV.    THE   MASS-ACTION   LAW   FOR   REACTIONS   INVOLVING   SOLID   PHASES 

109.  Form  of  the  Mass-Action  Expression    .  ^  .  '  .  '  ;.     *     .     .      .  165 
no.   Reactions  Involving  Solid  and  Gaseous  Substances    ....  165 
in.   Solubility  of  Unionized  and  Slightly  Ionized  Substances       .      .  166 

112.  Solubility  of  Largely  Ionized  Substances 167 

113.  The  Mass-Action  of  Imperfect  Solutes.    The  Concept  of  Activity  169 

CHAPTER  VIII:    EQUILIBRIUM   OF   CHEMICAL   SYSTEMS 
IN   RELATION  TO  THE   PHASES  PRESENT 

I.  GENERAL   CONSIDERATIONS 

114.  General  Considerations 174 

II.  ONE-COMPONENT   SYSTEMS 

115.  Representation  of  the  Equilibrium-Conditions  by  Diagrams      .  176 

III.    THE   PHASE   RULE 

116.  The   Concept  of  Variance  and  Inductive  Derivation  of  the 

Phase  Rule r8o 

117.  Discussion  of  the  Concept  of  Components 181 

1 1 8.  Derivation  of  the  Phase  Rule  from  the  Perpetual-Motion  Principle  183 

IV.    TWO-COMPONENT   SYSTEMS 

119.  Systems  with  Solid  and  Gaseous  Phases ^7 

120.  Systems  with  Solid,  Liquid,  and  Gaseous  Phases.     Pressure- 

Temperature  Diagrams j88 

121.  Systems  with  Solid  and  Liquid  Phases.    Temperature-Compo- 

sition Diagrams I9o 


CONTENTS  xv 

PAGE 

122.  Systems  with  Liquid  and  Gaseous  Phases 197 

123.  Systems  Involving  Solid  Solutions 197 


V.    THREE-COMPONENT   SYSTEMS 


124.  Systems  with  Liquid  and  Solid  Phases.    Temperature-Composi- 

tion Diagrams 201 

125.  Systems  with  Gaseous,  Liquid,  and  Solid  Phases  in  Relation  to 

the  Phase  Rule  and  Mass-Action  Law 203 


PART  III 

THE  ENERGY  EFFECTS  ATTENDING  CHEMICAL  CHANGES, 

AND  THE  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

IN  RELATION  TO  THESE  EFFECTS 

CHAPTER  IX:     THE  PRODUCTION  OF  HEAT 
BY  CHEMICAL  CHANGES 

I.    THE  FUNDAMENTAL  PRINCIPLES   OF  THERMOCHEMISTRY 

126.  The  First  Law  of  Thermodynamics       .     .     -. 207 

127.  Heat-Effects  Attending  Isothermal  Changes  in  State       .     .     .  208 

128.  Change  in  Energy-Content  and  in  Heat-Content      . .     .     .     .  209 

129.  Expression  of  Changes  in  Heat-Content  by  Thermochemical 

Equations •    .  211 

130.  Indirect  Determination  of  the  Heat-Effects  of  Chemical  Changes    214 

131.  Influence  of  Temperature  on  the  Heat-Effects  Attending  Chem- 

ical Changes       .     '.     ..,'...    ..........     215 

H.    GENERAL  RESULTS  OF  THERMOCHEMICAL  INVESTIGATIONS 

132.  Heat-Effects  Attending  Changes  in  the  State  of  Aggregation  of 

Substances 217 

133.  Heats  of  Reaction  in  Aqueous  Solution 218 

134.  Applications  of  Thermochemical  Principles 221 

CHAPTER  X:   THE  PRODUCTION  OF  WORK  BY  ISOTHERMAL 

CHEMICAL   CHANGES  IN  RELATION  TO  THEIR 

EQUILIBRIUM  CONDITIONS 

I.    THE  SECOND  LAW  OF  THERMODYNAMICS  AND  THE  CONCEPT  OF  FREE-ENERGY 

135.  The  Second  Law  of  Thermodynamics  and  its  Application  to  Iso- 

thermal Changes  in  State 223 

136.  The  Concepts  of  Work-Content  and  Free-Energy      ....     226 


xvi  CONTENTS 

II.    FREE-ENERGY  CHANGES  ATTENDING  PHYSICAL  CHANGES 

PAGE 

137.  Free-Energy  Changes  Attending  Changes  in  Volume  and  Pres- 

sure  230 

138.  Free-Energy  Changes  Attending  the  Transfer  of  Substances  be- 

tween Solutions  of  Different  Concentrations 231 

139.  Free-Energy  Changes  Attending  the  Transfer  of  Substances  be- 

tween Different  Phases 234 

m.    FREE-ENERGY   CHANGES   ATTENDING   CHEMICAL   CHANGES 

140.  The   Free-Energy   Equation   for   Chemical   Changes   between 

Perfect  Gases  *-*     . 238 

141.  The   Free-Energy   Equation   for   Chemical   Changes   between 

Perfect  Solutes 240 

142.  The  Free-Energy  Equations  for  Chemical  Changes  between 

Solid  Substances  and  Perfect  Gases  or  Perfect  Solutes      .     .     242 


CHAPTER  XI:  THE  PRODUCTION  OF  WORK  FROM  ISOTHER- 
MAL CHANGES  BY  ELECTROCHEMICAL  PROCESSES 

I.    CHANGES  IN  STATE  AND  IN  FREE  ENERGY  IN  VOLTAIC  CELLS 

143.  Introduction 243 

144.  Changes  in  State  in  Voltaic  Cells 243 

145.  The  Production  of  Work  and  the  Corresponding  Decrease  of 

Free  Energy  in  Voltaic  Cells  .      .      ...     ...     .     .     .     247 

H.     THE   ELECTROMOTIVE   FORCE   OF  CELLS   UNDERGOING   ONLY  CHANGES 
IN   CONCENTRATION  "OR  PRESSURE 

146.  Change  of  the  Electromotive  Force  of  Voltaic  Cells  with  the 

Concentration  of  the  Solutions 249 

147.  The  Electromotive  Force  of  Concentration-Cells  .     .     .     .     .     250 

HI.    THE  ELECTROMOTIVE  FORCE  OF  CELLS  UNDERGOING  CHEMICAL  CHANGES 
AND   ITS   RELATION  TO  THEIR  EQUILIBRIUM   CONDITIONS 

148.  Chemical  Changes  in  Voltaic  Cells 252 

149.  The  Electromotive  Force  of  Cells  in  Relation  to  the  Equilibrium 

Conditions  of  the  Chemical  Reactions    .     .     .     ...     .     252 

IV.    ELECTRODE-POTENTIALS   AND   LIQUID-POTENTIALS 

150.  The  Nature  of  Electrode-Potentials '.     .     .     254 

151.  The  Expression  of  Electrode-Potentials      .......     255 

152.  Change  of  Electrode-Potentials  with  the  Ion-Concentrations, 

and  the  Concept  of  Molal  Electrode-Potentials      ....     256 


CONTENTS  xvii 

PAGE 

153.  Values  of  the  Molal  Electrode-Potentials 259 

154.  Derivation  of  Related  Molal  Electrode-Potentials  from  One 

Another 260 

155.  The  Electromotive  Force  of  Cells  with  Dilute  Solutions  in  Re- 

lation to  the  Molal  Electrode-Potentials 260 

156.  The  Electromotive  Force  of  Cells  with  Concentrated  Solutions  .     261 

157.  The  Nature  of  Liquid-Potentials  and  Expressions  for  their  Ap- 

proximate Evaluation .     262 

158.  Derivation  of  the  Liquid-Potential  Equations 264 

159.  Determination  of  Ion-Concentrations  and  of  Equilibrium-Con- 

stants by  Means  of  Electromotive-Force  Measurements    .     .     266 

V.     THE  EQUILIBRIUM  OF  OXIDATION  REACTIONS  IN  RELATION 
TO  THE  ELECTRODE-POTENTIALS 

160.  Derivation  of  the  Equilibrium-Constants  of  Oxidation  Reac- 

tions from  the  Molal  Electrode-Potentials 269 

VI.    VOLTAIC  ACTION,   ELECTROLYSIS,   AND  POLARIZATION 

161.  Concentration-Changes  Attending  Voltaic  Action  and  the  Re- 

sulting Polarization 271 

162.  Electrolysis  in  Relation  to  Minimum  Decomposition-Potential    272 

163.  Electrolysis  in  Relation  to  Polarization      .     .     .     ...     .     .     273 

CHAPTER   XII:     THE   EFFECT   OF   TEMPERATURE   ON   THE 

WORK  PRODUCIBLE  BY  ISOTHERMAL  CHEMICAL  CHANGES 

AND  ON  THEIR  EQUILIBRIUM   CONDITIONS 

I.    THE   FUNDAMENTAL   SECOND-LAW  EQUATION 

164.  The  Quantity  of  Work  Producible  from  a  Quantity  of  Heat  that 

Passes  from  One  Temperature  to  Another    .     „     .     .     .     .     277 

H.     THE    EFFECT    OF   TEMPERATURE    ON   THE    EQUILIBRIUM 
OF   UNIVARIANT    SYSTEMS 

165.  The  Effect  of  Temperature  on  the  Pressure  at  which  the  Phases 

of  Univariant  Systems  are  in  Equilibrium 280 

HI.     THE  EFFECT  OF  TEMPERATURE  ON  THE  EQUILIBRIUM 
OF  CHEMICAL  REACTIONS   IN   GENERAL 

1 66.  The   Effect   of   Temperature   on   the   Free-Energy   Decrease 

Attending  Any  Isothermal  Change  in  State 284 

167.  The  Effect  of  Temperature  on  the  Equilibrium  of  Chemical 

Changes  Involving  Perfect  Gases       ........     285 

1 68.  The  Effect  of  Temperature  on  the  Equilibrium  of  Chemical 

Changes  Involving  Perfect  Solutes    ....     .   /.     .     .     .     288 


xviii  CONTENTS 

IV.  THE  EFFECT  OF  TEMPERATURE  ON  THE  ELECTROMOTIVE  FORCE 
OF  VOLTAIC  CELLS 

PAGE 

169.  The  Effect  of  Temperature  on  the  Electromotive  Force  of  Voltaic 

Cells 290 


CHAPTER  XIII:   SYSTEMIZATION  OF  FREE-ENERGY  VALUES 

170.  Importance  of  Systemizing  Free-Energy  Values 292 

171.  Expression  of  Free-Energy  Changes  by  Equations      ....  292 

172.  Determination  of  the  Free-Energies  of  Substances      ....  294 

173.  The  Equilibrium-Conditions  of   Chemical  Reactions  Derived 

from  the  Free-Energies  of  the  Substances  Involved      .     .     .     295 


NOTATION  AND   INDEX 

Notation     .     .     .     .     .     .     .     .     .     .     ; 297 

Index 299 


PART   I 

THE  ATOMIC,  MOLECULAR,  AND  IONIC  THEORIES 
AND  THE  PROPERTIES  OF  SUBSTANCES  DI- 
RECTLY RELATED  TO  THESE  THEORIES 


CHAPTER  I 

THE   COMPOSITION  OF   SUBSTANCES  AND   THE 
ATOMIC   THEORY 


1.  The  Field  of  Chemistry.     Its  General  Principles  the  Subject  of 
this  Course.  —  Chemistry  treats  of  the  composition  and  molecular 
structure  of  substances,  of  their  properties  in  relation  to  their  composi- 
tion and  molecular  structure,  of  changes  in  their  composition,  and  of 
the  effects  attending  such  changes.     Physics  treats  of  the  properties 
of  substances  without  relation  to  their  composition,  and  of  changes  in 
state  that  do  not  involve  changes  in  composition. 

This  course  on  Chemical  Principles  is  devoted  to  a  consideration  of 
the  more  important  general  principles  which  express  the  properties  and 
reactions  of  chemical  substances.  The  subject  here  considered  is 
sometimes  called  general,  theoretical,  or  physical  chemistry.  The  last 
of  these  names,  which  is  now  most  commonly  employed,  carries  the 
unfortunate  implications  that  the  subject  is  a  separate  branch  of 
chemistry  coordinate  with  inorganic,  organic,  analytical,  or  applied 
chemistry,  and  that  it  deals  mainly  with  the  physical  sides  of 
chemical  science;  whereas,  in  reality,  it  deals  with  the  principles 
common  to  all  branches  of  chemistry  and  is  concerned  primarily  with 
purely  chemical  phenomena,  such  as  the  properties  of  substances  in 
relation  to  their  molecular  structure,  and  such  as  the  rate  and 
equilibrium  of  chemical  reactions  and  the  attendant  effects.  The 
general  plan  adopted  for  the  presentation  of  the  subject  in  this  book 
may  be  seen  by  reference  to  the  headings  of  the  Parts  and  Chapters 
in  the  Table  of  Contents  given  on  the  preceding  pages. 

2.  Pure  Substances  and  Mixtures,  Elementary  and    Compound 
Substances,  the  Elements,  and  the  Law  of  Definite  Proportions.  — 
Out  of  the  materials  occurring  in  nature  there  can  be  prepared  sub- 
stances which,  when  subjected  to  suitable  processes  of  fractionation 
(that  is,  to  operations  which  resolve  the  materials  into  parts  or  frac- 
tions), always  yield  fractions  whose  properties  are  identical  when 
measured  at  the  same  pressure  and  temperature.     Such  substances  are 
called  pure  substances;  other  substances  which  can  be  resolved  by  such 
processes  into  fractions  with  different  properties  being  called  mixtures. 
For  example,  whether  a  solid  material  is  a  pure  substance  or  mixture 

3 


COMPOSITION  OF  SUBSTANCES 


may  be  determined  by  partially  melting  or  vaporizing  it  or  by  partially 
dissolving  it  in  solvents,  and  by  comparing  the  value  of  the  density, 
melting-point,  or  some  other  sensitive  property,  of  the  unmelted,  un- 
vaporized,  or  undissolved  part  with  that  of  the  original  material. 

The  fundamental  idea  involved  in  the  preceding  considerations  is 
that  there  exists  an  order  of  substances,  called  pure  substances,  of  rela- 
tively great  stability  toward  resolving  agencies,  each  one  of  which  has 
a  perfectly  definite  set  of  properties,  sharply  differentiated  from  those 
of  other  pure  substances;  so  that  there  is  not  a  continuous  series  of 
pure  substances  whose  properties  pass  over  into  one  another  by  insensi- 
ble gradations. 

Nearly  all  pure  substances,  called  compound  substances  because  of 
the  behavior  to  be  now  mentioned,  can  be  converted,  by  subjecting 
them  to  sufficiently  powerful  resolving  conditions,  into  a  small  number 
of  other  pure  substances,  called  elementary  substances,  which  are  not 
further  decomposed  by  chemical  or  physical  processes.  In  corre- 
spondence with  this  fact  all  substances  are  considered  to  be  composed 
of  a  small  number  of  kinds  of  matter,  about  ninety  in  all,  called  the 
dements;  each  elementary  substance  consisting  of  only  one  of  these 
elements,  and  each  compound  substance  of  two  or  more  of  them. 

Although  some  elements,  those  contained  in  the  so-called  radio- 
active substances,  have  been  found  to  be  undergoing  spontaneous 
disintegration,  elementary  substances  obviously  constitute  a  separate 
order  of  substances  of  very  great,  but  not  unlimited,  stability  toward 
decomposing  agents. 

The  general  principle  of  the  definiteness  of  the  properties  of  pure 
substances  applies  also  to  the  elementary  composition  of  pure  sub- 
stances. This  fact  is  expressed  by  the  law  of  definite  proportions,  which 
states  that  a  pure  substance,  however  it  be  prepared,  always  contains 
its  elements  in  exactly  the  same  proportions  by  weight. 

3.  The  Law  of  Combining  Weights.  —  To  the  various  elements 
definite  numerical  values  can  be  assigned  which  when  multiplied  or 
divided  by  small  whole  numbers  accurately  express  the  weights  of  the 
elements  which  are  combined  with  one  another  in  all  pure  substances^ 
Such  numerical  values  are  called  the  combining  weights  of  the  elements. 
Adopting  1 6  as  the  combining  weight  of  oxygen,  the  combining  weight 
of  any  other  element  may  be  defined  to  be  that  weight  of  it  which 
combines  with  16  parts  of  oxygen,  or  with  a  quantity  of  oxygen  (such 
as  8,  24,  or  32  parts)  which  stands  to  16  in  the  ratio  of  small  whole 


THE  ATOMIC  THEORY  5 

numbers.  It  will  be  seen  from  this  definition  that  a  combining  weight 
is  an  arbitrary  multiple  of  an  exact  quantity  ^  The  principle  that  the 
composition  of  all  pure  substances  can  be  accurately  expressed  in 
terms  of  the  so-defined  combining  weights  of  the  elements  is  called  the 
law  of  combining  weights. 

Prob.  i.  —  The  Concept  and  Law  of  Combining  Weights.  —  a.  Two 
oxides  of  sulfur  are  found  by  analysis  to  contain  50.05%  and  40.04% 
sulfur.  Derive  from  these  analytical  results  two  values  for  the  com- 
bining weight  of  sulfur,  and  show  that  these  values  stand  to  one 
another  in  the  ratio  of  small  whole  numbers,  b.  Derive  a  combining 
weight  for  iron  from  the  fact  that  one  of  its  oxides  contains  30.06%  of 
oxygen,  c.  Derive  another  value  for  the  combining*  weight  of  iron 
from  the  fact  that  one  of  its  sulfides  contains  36.47%  of  sulfur,  whose 
combining  weight  was  found  in  a\  and  show  that  the  two  values  of  the 
combining  weight  of  iron  stand  to  each  other  in  the  ratio  of  small 
whole  numbers. 

4.  The  Atomic  Theory.  —  The  fact  that  elements  combine  with 
one  another  only  in  the  proportions  of  their  combining  weights  or 
multiples  of  them  originally  suggested  the  following  fundamental 
atomic  and  molecular  hypotheses: 

(1)  The  matter  constituting  each  element  is  made  up  of  extremely 
small  particles,  called  atoms,  which  are  alike  in  every  respect,  and 
which  are  not  subdivided  hi  ordinary  chemical  or  physical  changes; 
there  being  as  many  different  kinds  of  atoms  as  there  are  elements. 

(2)  The  atoms  associate  with  one  another,  usually  in  small  numbers, 
forming  a  new  order  of  distinct  particles,  called  molecules.     Pure  sub- 
stances usually  consist  of  only  one  kind  of  molecule,  while  mixtures 
always  contain  two  or  more  kinds;  and  the  molecules  of  elementary 
substances  consist  of  atoms  of  the  same  kind,  while  those  of  compound 
substances  consist  of  atoms  of  different  kinds. 

By  supplementing  these  hypotheses  with  certain  other  hypotheses, 
developing  them  to  their  logical  conclusions,  and  applying  these  con- 
clusions to  physical  and  chemical  phenomena,  many  of  the  physical 
properties  and  chemical  reactions  of  substances  have  been  explained. 
Indeed,  the  adequacy  of  the  fundamental  hypotheses  has  now  been 
confirmed  in  so  many  ways  that  there  is  no  longer  any  doubt  that 
substances  are  really  made  up  of  atoms  and  molecules  of  substantially 
the  assumed  character. 

These  hypotheses  and  the  conclusions  drawn  from  them  constitute 
the  atomic  and  molecular  theories;  the  former  term  being  commonly 


6  THE  COMPOSITION  OF  SUBSTANCES 

used  when  properties  that  are  more  directly  related  to  the  characteris- 
tics of  the  atoms  or  to  the  atomic  structure  of  the  molecule  are  under 
consideration;  the  latter  term  when  properties  that  are  determined 
primarily  by  the  characteristics  of  the  molecule  as  a  whole  are  being 
studied. 

A  substance  which  contains  only  a  single  kind  of  molecule  is  called 
a  chemical  substance  (or  by  some  authors  a  molecular  species).  Pure 
substances  normally  consist  of  single  chemical  substances;  but  they 
may  consist  of  two  or  more  chemical  substances  which  are  capable  of 
conversion  into  one  another  and  which  are  always  present  in  definite 
proportions  at  any  given  temperature  and  pressure  in  consequence  of 
the  rapid  establishment  of  equilibrium  between  them.  Thus,  the 
pure  substance  liquid  water  contains  the  two  chemical  substances 
whose  molecules  are  H2O  and  H4O2,  these  being  always  present  in 
definite  proportions  at  any  given  temperature  and  pressure,  since 
equilibrium  is  almost  instantaneously  established  between  them;  but 
the  pure  substance  water- vapor  at  small  pressures  is  a  single  chemical 
substance,  for  it  contains  only  molecules  of  the  form  H^O.  Pure 
substances  which,  like  water,  water-vapor;  and  ice,  are  converted  into 
one  another  by  changes  of  pressure  and  temperature,  are  commonly 
spoken  of  as  the  same  substance;  but  they  may  consist  of  different 
chemical  substances,  as  has  just  been  illustrated. 

The  relative  weights  of  the  atoms  of  the  various  elements  and  of  the 
molecules  of  the  various  substances  are  called  their  atomic  weights  and 
molecular  weights,  respectively;  and  as  a  standard  of  reference,  the 
weight  of  the  oxygen  atom  taken  as  1 6  is  adopted. 

The  atomic  theory  evidently  requires  that  the  weights  of  elements 
that  combine  with  one  another  be  proportional  to  the  weights  of  their 
atoms  or  to  multiples  of  those  weights;  in  other  words,  that  the  atomic 
weights  be  equal  to  the  combining  weights  or  to  multiples  of  them. 

5.  Determination  of  Combining  and  Atomic  Weights.  —  The 
determination  of  the  atomic  weight  of  an  element  involves  two  distinct 
steps:  (i)  the  determination  of  its  combining  weight;  and  (2)  the 
determination  of  that  multiple  of  its  combining  weight  which  is  its 
atomic  weight. 

The  combining  weight  of  an  element  is  experimentally  determined 
by  preparing  one  of  its  compounds  or  the  elementary  substance  itself 
in  a  state  of  the  highest  possible  purity,  and  completely  converting  it 
into  another  pure  substance,  the  other  elements  contained  in  the  sub- 


THE  ATOMIC  THEORY  7 

stances  being  of  known  combining  weights.  The  quantities  of  the 
initial  substance  taken  and  of  the  final  substance  obtained  are  accu- 
rately weighed,  and  from  the  ratio  of  these  weights  the  combining 
weight  is  calculated.  For  example,  the  combining  weight  of  hydrogen 
has  been  determined  by  burning  a  weighed  quantity  of  pure  hydrogen 
gas  with  oxygen  and  weighing  the  water  formed;  also  by  finding 
the  weight  of  the  oxygen  consumed.  Values  for  the  weight  ratios 
H2 :  H2O  and  H2 :  O  were  thus  established,  and  from  each  of  them  the 
combining  weight  of  hydrogen  was  computed.  The  following  prob- 
lem illustrates  the  methods  of  determining  combining  weights  in 
general,  and  it  shows  some  of  the  fundamental  determinations  on 
which  our  present  system  of  atomic  weights  is  based. 

Prob.  2.  —  Determination  of  Combining  Weights.  —  Determine  the 
exact  combining  weights  of  silver,  potassium,  and  chlorine  from  the 
following  data:  In  a  series  of  eight  experiments  801.48  g.  of  pure 
potassium  chlorate  were  ignited  or  treated  with  hydrochloric  acid, 
yielding  487.66  g.  of  potassium  chloride.  In  another  series  of  five 
experiments  24.452  g.  of  pure  potassium  chloride  were  dissolved  in 
water  and  precipitated  with  silver  nitrate,  whereby  47.013  g.  of  silver 
chloride  were  obtained.  In  a  third  series  of  ten  experiments  82.669  g. 
of  pure  silver  were  dissolved  in  nitric  acid  and  precipitated  with  hydro- 
chloric acid,  yielding  109.840  g.  of  silver  chloride. 

Ans.  Ag,  35.978  x;  Cl,  11.825  y\  K,  13.038  z;  where  x,  y,  and  z  are 
ratios  of  whole  numbers  not  determinable  by  gravimetric  methods. 

The  multiple  of  the  analytically  determined  combining  weight 
which  is  the  most  probable  atomic  weight  of  each  element  was  origi- 
nally derived  for  most  of  the  elements  from  theoretical  principles  relat- 
ing to  certain  physical  properties,  especially  the  pressure-volume 
relations  of  gaseous  compounds,  the  heat-capacities  of  the  elementary 
substances  in  the  solid  and  gaseous  states,  and  the  isomorphism  of 
corresponding  compounds  of  different  elements  in  the  crystalline 
state.  The  principles  relating  to  the  first  two  of  these  properties  and 
their  applications  to  atomic  weight  determinations  are  considered  in 
Chapters  II  and  IV.  The  system  of  atomic  weights  so  derived  finds 
its  chief  justification,  however,  in  the  fact  that  it  has  led  to  molecular 
formulas  of  chemical  substances  which  represent  their  chemical 
behavior  in  a  remarkably  satisfactory  manner.  It  has  also  led  to  the 
formulation  of  the  periodic  system  of  the  elements,  which  affords  a 
qualitative  representation  of  many  of  the  physical  and  chemical 
properties  of  elementary  and  compound  substances  as  a  progressively 
changing  and  periodically  recurring  function  of  the  atomic  weights. 


8 


THE  COMPOSITION  OF  SUBSTANCES 


6.  Values  of  the  Atomic  Weights.  —  The  existing  data  and  all 
newly  determined  data  relating  to  atomic  weights  are  critically  con- 
sidered at  frequent  intervals  by  an  International  Committee  on 
Atomic  Weights,  and  a  table  is  published  each  year  showing  the  most 
probable  values.  The  following  table  contains  the  latest  values 
adopted  for  the  more  important  elements.  The  values  are  given  to 
such  a  number  of  decimal  places  that  the  last  one  is  probably  in  error 
by  not  more  than  one  or  two  units. 

VALUES  OF  THE  ATOMIC  WEIGHTS 


Aluminum 

.     Al     27.0 

Lead  .     . 

.     Pb    207.20 

Antimony 

.     Sb   120.2 

Lithium  . 

.     Li        6.94 

Argon  . 

.     A      39.9 

Magnesium 

.     Mg     24.32 

Arsenic      : 

.     As     74.96 

Manganese 

.     Mn     54.93 

Barium      .' 

.     Ba  137.37 

Mercury 

.     Hg    200.6 

Beryllium 

.     Be      9.1 

Neon 

.     Ne      20.2 

Bismuth    .  > 

.     Bi    209.0 

Nickel     . 

.     Ni      58.68 

Boron  . 

.     B       10.9 

Nitrogen 

.     N       14.008 

Bromine    . 

.     Br     79.92 

Oxygen    . 

.     O        16.00 

Cadmium 

Cd  112.40 

Phosphorus 

.     P       31.04 

Calcium    . 

.     Ca    40.07 

Platinum 

.     Pt     195.2 

Carbon 

C       12.005 

Potassium 

.     K       39.10 

Chlorine    . 

.     Cl     3546 

Radium 

.     Ra    226.0 

Chromium 

.     Cr     52.0 

Silicon     . 

.     Si       28.1 

Cobalt       . 

.     Co     58.97 

Silver 

.     Ag    107.88 

Copper 

.     Cu    63.57 

Sodium    . 

.     Na     23.00 

Fluorine    . 

.     F       19.0 

Strontium 

.     Sr       87.63 

Gold    .      . 

.     Au  197.2 

Sulfur      . 

.     S        32.06 

Helium 

.     He      4.00 

Thallium 

.     Tl     204.0 

Hydrogen 

.     H        1.008 

Tin    .      . 

.     Sn    118.7 

Iodine 

.     I      126.92 

Titanium 

.     Ti      48.1 

Iron     .      . 

.     Fe     55-84 

Zinc  .      . 

.     Zn      65.37 

7.  Chemical  Formulas,  Formula- Weights,  and  Equivalent- Weights.  — 
In  order  to  express  the  gravimetric  composition  of  compounds,  the 
symbols  of  the  elements  are  considered  to  represent  their  atomic 
weights  and  are  written  in  sequence  with  such  integers  as  subscripts 
as  will  make  the  resulting  formula  express  the  proportions  by  weight 
of  the  elements  in  the  compound. 

The  formula  is  commonly  so  written  as  to  represent  also  the  number 
of  atoms  of  each  element  present  in  the  molecule. 

The  formula  represents,  in  addition,  a  definite  weight  of  the  substance, 
namely  the  weight  in  grams  which  is  equal  to  the  sum  of  the  numbers 
represented  by  the  symbols  of  the  elements  in  the  formulas.  This 
weight  is  called  the  formula-weight  of  the  substance.  Thus  the  formula 
HNOs  denotes  1.0084-14.01+ (3X16.00)  or  63.02  grams  of  nitric  acid. 


THE  ATOMIC  THEORY  g 

Those  weights  of  various  substances  which  enter  into  chemical 
reactions  with  one  another  are  called  equivalent-weights.  Adopting 
one  formula- weight  or  i.ooS  grams  of  the  element  hydrogen  as  the 
standard  of  reference,  the  equivalent-weight  or  one  equivalent  of  any 
substance  is  denned  to  be  that  weight  of  it  which  reacts  with  this 
standard  weight  of  hydrogen,  or  with  that  weight  of  any  other  sub- 
stance which  itself  reacts  with  this  standard  weight  of  hydrogen. 
Thus  the  equivalent-weight  of  each  of  the  following  substances  is 
that  fraction  of  its  formula-weight  which  is  indicated  by  the  coefficient 
preceding  the  formula:  jCfcjiOs;  iAg;jZn;  Jfii;  iNaOH;  jBa(OH)2; 
JH2SO4;  iH3PO4;  f  A1C13;  iK4Fe(CN)6.  The  equivalent-weight  of  a 
substance  may  have  different  values  depending  on  whether  it  is 
considered  with  reference  to  a  reaction  of  metathesis  or  to  one  of 
oxidation  and  reduction.  Thus,  the  metathetical  equivalent  of  ferric 
chloride  is  jFed3,  but  its  oxidation-equivalent  (with  respect  to  its 
conversion  to  ferrous  chloride)  is  iFed3;  the  metathetical  equivalent  of 
potassium  chlorate  is  iKC!O3,  but  its  oxidation-equivalent  (with 
reference  to  its  reduction  to  KC1)  is  £KC103. 


CHAPTER  II 

THE  MOLAL  PROPERTIES  OF  GASES  AND  THE   MOLEC- 
ULAR AND   KINETIC   THEORIES 


I.    THE  VOLUME  OF  GASES  IN  RELATION  TO  PRESSURE,  TEMPERATURE, 
AND  MOLAL  COMPOSITION 

8.  The  Volume  of  Perfect  Gases  in  Relation  to  Pressure  and 
Temperature.  —  Properties  whose  magnitudes  are  determined  pri- 
marily by  the  number,  and  not  by  the  nature,  of  the  molecules  present 
are  called  mold  properties.  The  properties  of  this  kind  exhibited  by 
gas&Stre  considered  in  this  chapter. 

The  properties  of  gases,  in  general,  conform  more  and  more  closely 
to  certain  limiting  laws,  known  as  the  laws  of  perfect  gases,  as  then- 
pressure  approaches-  zero.  The  principles  relating  to  the  effects  of 
pressure  and  temperature  on  the  volume  of  perfect  gases  will  be  first 
considered. 

At  any  definite  temperature  the  pressure  p  of  a  definite  weight  m 
of  any  perfect  gas  is  inversely  proportional  to  its  volume  v;  or  its 
pressure- volume  product  p  v  is  independent  of  its  pressure  or  volume. 
Since  density  (d)  is  defined  to  be  the  ratio  (m/v)  of  the  weight  of  a  sub- 
stance to  its  volume,  this  law  may  als^be  stated  as  follows:  At  any 
definite  temperature  the  density  of  any  definite  perfect  gas  is  propor- 
tional to  its  pressure.  This  principle  is  known  as  Boyle's  law.  r 

The  pressure-volume  product  increases  with  rise  in  temperature  by 
the  same  fractional  amount  for  differei^  perfect  gases  when  they  are 
heated  from  the  same  initial  to  the  same  final  temperature.  For  ex- 
ample, the  pressure-volume  product  increases  by  36.6i2percent  of  its 
value  at  o°  when  the  temperature  becomes  100°,  whatever  be  the  nature 
of  the  perfect  gas.  This  law  is  known  as  the  law  of  temperature-effect. 
It  is  mathematically  expressed  by  the  equation  ff&»—piVi)/(piVi)~ 
f(/i,  AJ),  where  f(/t,  t^)  denotes  the  same  function  of  the  temperatures  ti 
and  AJ  whatever  be  the  nature  of  the  perfect  gas. 

The  form  of  the  functional  relation  between  the  pressure-volume 
product  and  the  temperature  evidently  depends  on  the  definition  of 
temperature  adopted;  and  this  relation  assumes  the  simplest  form 
when  the  concept  of  absolute  temperature  is  employed.  For  absolute 

10 


PRESSURE-VOLUME  RELATIONS  OF  GASES  n 

temperature  is  so  defined  that  it  is  directly  proportional  to  the  pressure- 
volume  product  of  a  perfect  gas;  that  is,  it  is  defined  by  the  equation 
T/To  =  (pv)/(poVo),  where  p0vQ  is  the  value  of  pv  at  any  specified 
temperature  TO,  which  for  definiteness  may  here  be  taken  as  that  of 
ice  when  melting  under  a  pressure  of  one  atmosphere.  The  value  of 
To  is  dependent  on  the  temperature-scale  adopted.  We  may  deter- 
mine its  value  on  the  centigrade  scale  from  the  following  considera- 
tions. This  scale  is  based  on  the  convention  that  there  shall  be  a 
difference  of  100  degrees  between  the  temperature  of  melting  ice  and 
the  temperature  of  boiling  water,  when  both  are  at  a  pressure  of  one 
atmosphere.  Now  by  writing  the  precedinJgjnuation  in  the  form 
(T-TQ)/TO  =  (PV-POVQ)/PQVQ,  and  substifc^  •Lj'o  the  value 
100  and  for  the  second  member  the  quantil  >  givoyi^he 

preceding  paragraph,  we  find  the  value  273.14°  for  To,  the  temperature 
of  melting  ice  at  a  pressure  of  i,.ajm.  on  the  absolute^£gp|?ade  gfe. 
Since  this  temperature  is  calleet'p?  .<|Byj|^|^^^cfentigrade  scale, 
and  since  this  scale,  like  the  absolute  scale,  is  1  Hn  the  convention 
of  proportionality  between  increase  of  temperature  and  the  increase 
in  the  pressure- volume  product  of  a  perfect  gas^he  absolute  centigrade 
temperature  T  is  always  equal  to  the  ordinary  centigrade  temperature 
/  increased  by  273.14,  or  approximately  273;  black-face  type  being 
used  here  and  throughout  the  book  to  show  approximate  values, 
given  with  an  accuracy  of  o.i  —  0.2  percent,  of  fundamental  con- 
stants which  it  is  well  to  rem(|nber. 

The  laws  above  stated,  the  definition  of  absolute  temperature,  and 
the  obviojp,  proportionality  between  the  value  of  the  pressure- volume 
product  and  the  weight  m  of  the  gas  are  all  expressed  by  the  equation 
pv=mRT,  in  which  R  is  a  cogitant,  evidently  representing  the  value 
otpv/T  for  qj^gram  of  the  gas,  which  has  different  values  for  different 

$P 

Prob  i.  — Derivation  from  Experimental  Data  of  pv/T  for  One  Gram 
of  Carbon  Dioxty* —  a.  From  the  fact  that  the  density  of  carbon  dioxide 
is  0.0019765  g.  per  ccm.  at  o°  and  i  atm.,  calculate  the  exact  value  of 
p  v/T  under  these  conditions  for  one  gram  of  the  gas.  b.  Taking  into 
account  the  fact  that  at  this  temperature  and  pressure  the  density  has 
been  found,  by  extrapolating  from  a  series  of  measurements  made  at 
pressures  between  o.i  and  i  atm.,  to  be  0.68%  greater  than  that  re- 
quired by  the  perfect -gas  law,  calculate  for  this  gas  the  value  of  the 
constant  R,  the  pressure  being  expressed  in  atmospheres  and  the  volume 
in  cubic  centimeters.  Ans.  b,  1.8649. 


12  THE  MOLAL  PROPERTIES  OF  GASES 

9.  Law  of  Combining  Volumes  and  the  Principle  of  Avogadro.  — 

The  foregoing  physical  laws  acquire  an  important  chemical  significance 
by  reason  of  the  law  of  combining  volumes,  which  may  be  stated  as 
follows:  Those  quantities  of  perfect  gases  that  are  involved  in  chem- 
ical reactions  with  one  another  have,  at  the  same  temperature  and 
pressure  volumes,  which  are  equal  or  small  multiples  of  one  another. 
Thus  the  quantities  of  hydrogen  and  of  oxygen  which  unite  with  each 
other  to  form  water  have  volumes  whose  ratio  approaches  the  exact 
value  2 :  i  as  the  pressure  of  the  gases  approaches  zero.  In  other 
words,  those  quantities  of  different  substances  which  as  perfect  gases 
have  the  same  value  of  the  product  pv/T  are  equal  to,  or  are  small 
multiples  of,  the  quantities  which  are  involved  in  chemical  reactions 
with  one  another.  Now,  since  according  to  the  molecular  theory 
substances  react  by  molecules,  these  quantities  of  different  perfect 
gases  which  have  the  same  value  of  p  v/  T  must  contain  either  an  equal 
number  of  molecules  or  small  multiples  of  an  equal  number. 

These  considerations  suggest  a  simpler  principle,  known  as  the 
principle  of  Avogadro,  which  may  be  stated  as  follows:  Those^  quanti- 
ties of  different  perfect  gases  which  have  the  same  volume  at  the  same 
temperature  and  pressure,  and  therefore  the  same  value  oipv/T  at  any 
temperature  and  pressure,  contain  an  equal  number  of  molecules. 
The  densities  of  different  perfect  gases  at  the  same  temperature  and 
pressure  are  therefore  proportional  to  the  weights  of  their  molecules. 

This  principle,  originally  hypothetical,  has  now  been  so  fully  verified 
that  it  has  become  one  of  the  fundamental  laws  of  chemistry. 

10.  Empirical  Definition  of  Molecular  Weight  and  of  Mol.  —  The 
principle  of  Avogadro  evidently  enables  the  relative  weights  of  the 
molecules  of  different  gaseous  substances  to  be  determined.    To  ex- 
press these  relative  weights  more  definitely,  the  ratio  of  the  weight  of 
the  molecule  of  any  substance  to  the  weight  of  the  molecule  of  oxygen 
taken  as  32  is  commonly  considered,  this  ratio  being  called  the  molecu- 
lar weight  M  of  the  substance.    The  number  32  is  adopted  as  the  refer- 
ence quantity  of  oxygen,  since  (as  shown  in  Prob.  15)  it  corresponds 
to  the  adoption  of  1 6  as  the  atomic  weight  of  oxygen. 

The  so-defined  molecular  weight  of  a  substance  may*  evidently  be 
experimentally  determined  by  finding  the  number  of  grams  of  it  which 
have  that  value  oi  pv/T  which  32  grams  of  oxygen  have,  when  both 
substances  are  in  the  state  of  perfect  gases.  This  number  of  grams  is 
called  one  mol  or  one  molal  weight  of  the  substance. 


PRESSURE-VOLUME  RELATIONS  OF  GASES  13 

The  mol  is  a  unit  of  great  importance  in  chemical  considerations, 
both  because  it  is  directly  related  to  molecular  weight  and  because 
it  makes  possible  a  general  expression  of  the  laws  of  perfect  gases. 

11.  General  Expression  of  the  Laws  of  Perfect  Gases.  —  Repre- 
senting by  R  the  constant  value  of  p  v/  T  for  one  mol  and  by  ^V  the 
number  of  mols  of  the  gas  present,  the  laws  of  perfect  gases  may  be 
expressed  in  a  general  form  by  the  following  equation,  hereafter  called 
the  perfect-gas  equation: 


The  numerical  value  of  the  gas-constant  R  depends  on  the  units 
in  which  the  pressure,  volume,  and  temperature  are  expressed.  In 
scientific  work,  temperature  is  always  expressed  in  centigrade  degrees 
(here  on  the  absolute  scale);  volume  is  ordinarily  expressed  in  cubic 
centimeters  or  liters;  and  pressure  in  dynes  per  square  centimeter, 
millimeters  of  mercury,  or  atmospheres.  One  dyne  is  a  force  of  such 
magnitude  that  when  it  acts  on  a  freely  moving  mass  of  one  gram  it 
increases  its  velocity  each  second  by  one  centimeter  per  second;  and 
the  pressure  of  one  dyne  per  square  centimeter  is  called  one  bar,  io6 
bars  being  called  one  megabar.  One  atmosphere  is  a  pressure  equal 
to  that  exerted  by  a  column  of  mercury  76  cm.  in  height  at  o°  at  the 
sea-level  in  a  latitude  of  45°,  The  value  of  the  gas-constant  R  when 
the  pressure  is  in  atmospheres  and  the  volume  is  in  cubic  centimeters 
is  82.07  (approximately  82)»  Its  value  when  the  pressure  is  in  dynes 
per  square  centimeter  and  the  volume  is  in  cubic  centimeters  is 
8.3i6Xio7  ergs  per  degree.  Its  value  will  later  be  shown  to  be  1.985 
calories  per  degree. 

Prob*  2.  —  Value  of  the  Atmosphere.  —  Calculate  the  value  of  the 
atmosphere  in  dynes  per  sqcm.  and  in  megabars.  The  density  of 
mercury  at  o°  is  13.60.  The  force  of  gravity  acting  on  any  freely 
moving  body  increases  its  velocity  each  second  by  g  centimeters  per 
second.  .  The  value  of  g  at  the  sea-level  in  a  latitude  of  45°  is  980.6 
(980)  centimeters  per  second.  Ans.  1.013  Xio6  dynes  per  sqcm. 

Prob.  3.  —  Calculation  of  the  Gas-Constant.  —  a.  Calculate  precisely 
the  value  of  the  gas-constant  R  when  the  pressure  is  in  atmospheres 
and  the  volume  in  cubic  centimeters,  from  the  fact  that  one  gram  of 
oxygen  has  a  volume  of  7005  ccm.  at  o°  and  o.i  atm.,  under  which 
conditions  oxygen  does  not  deviate  appreciably  from  the  perfect- 
gas  law.  b.  Calculate  also  the  value  of  R  when  the  pressure  is  in 
dynes  per  square  centimeter,  and  the  volume  is  in  cubic  centimeters. 
c.  Calculate  the  volume  of  one  mol  of  a  perfect  gas  at  20°  and  i  atm. 
Ans.  c,  24060  ccm.  (approximately  24  1.) 


14  TEE  MOLAL  PROPERTIES  OF  GASES 

Applications  of  the  Perfect-Gas  Laws.  — 

Note.  —  In  the  following  problems,  and  in  general  throughout  the 
book,  assume,  unless  otherwise  indicated,  as  an  approximation  sufficient 
for  many  purposes,  that  gases  at  pressures  not  exceeding  i  atm.  con- 
form to  the  perfect-gas  equation. 

v,     Prob.  4.  —  Seven  grams  of  a  certain  gas  have  a  volume  of  6.35  1.  at 
^  20°  and  720  mm.    How  many  grams  make  one  mol? 
y    Prob.  5.  —  a.  What  is  the  volume  occupied  by  12  g.  of  ether  vapor 
/XC4Hi0O)  at  80°  and  600  mm.?    b.  What  is  the  density?    c.  What  is 
the  ratio  of  the  density  to  that  of  oxygen  at  the  same  temperature  and 
pressure? 

^   Prob.  6.  —  How  many  grams  of  iron  must  be  taken  to  produce  by  its 
action  on  sulfuric  acid  one  liter  of  hydrogen  (H2)  at  27°  and  i  atm.? 

It  will  be  seen  in  Art.  12  that,,  though  air  is  a  mixture,  it  has  in 
different  localities  a  fairly  constant  composition  and  that  the  pres- 
sures of  its  separate  constituents  are  determined  by  the  perfect-gas 
laws.  The  pressure  of  the  air  as  a  whole,  and  its  pressure-volume- 
temperature  relations  in  general,  can  therefore  be  simply  computed 
with  the  aid  of  the  perfect-gas  equation  by  regarding  air  as  a  single 
substance  having  a  molal  weight  equal  to  the  number  of  grams  of  air 
which  have  the  same  value  of  p  v/T  as  32  grams  of  oxygen.  The  so- 
defined  molal  weight  of  air  is  computed  from  the  most  accurate  density 
determinations  to  be  28.98  (approximately  29). 

Prob.  7.  —  Calculation  of  the  Molal  Weight  of  Air.  —  Calculate  the 
molal  weight  of  pure  dry  air  from  the  fact  that  its  density  is  0.0012930 
at  o°  under  a  pressure  of  760  mm.  of  mercury  at  sea-level  in  latitude  45°. 

12.  Dalton's  Law  of  Partial  Pressures.  —  The  (total)  pressure  p 
of  a  mixture  of  substances  in  the  gaseous  state  is  evidently  the  sum 
of  the  pressures  p\t  p2  . .  .  of  the  separate  substances.  That  is, 
p=p\-\-  p2  +•  • .  The  pressures  of  the  separate  substances  are  called 
the  partial  pressures. 

Partial  pressures  in  the  case  of  perfect  gases  may  be  calculated  as 
described  below.  They  cannot  be  directly  measured,  except  in  cases 
where  a  semipermeable  wall,  that  is,  a  wall  permeable  for  only  one  of 
the  substances  present  in  the  gas  mixture,  can  be  found.  Thus,  when 
a  platinum  vessel  containing  a  mixture  of  hydrogen  and  nitrogen  at 
a  high  temperature  is  immersed  in  an  atmosphere  of  hydrogen,  hydro- 
gen passes  through  the  platinum  walls  until  its  pressures  within  and 
without  the  vessel  become  equal;  the  difference  then  observed  between 
the  total  pressure  of  the  mixture  within  the  vessel  and  the  pressure 
of  the  hydrogen  outside  is  due  to  the  nitrogen,  which  does  not  pass 
through  the  wall;  and  this  difference  is  equal  to  its  partial  pressure. 


PRESSURE-VOLUME  RELATIONS  OF  GASES  15 

The  pressure  of  a  mixture  of  substances  in  the  state  of  a  perfect  gas 
is  found  to  be  equal  to  the  sum  of  the  partial  pressures  calculated  by 
the  principle  that  each  chemical  substance  (defined  as  in  Art.  4  to  be 
a  substance  consisting  of  molecules  of  one  particular  kind)  has  the 
same  pressure  as  it  would  if  it  were  alone  present  in  the  volume  occu- 
pied by  the  mixture.     This  principle,  which  is  of  great  importance  in 
chemical  considerations,  is  known  as  Dalton's  law  of  partial  pressures. 
/~SLn  accordance  with  this  law  and  the  perfect-gas  law,  the  partial 
/  pressures  of  the  separate  substances  are  evidently  given  by  the  expres- 
/sions,   pi  =  NiRT/v,  p2  =  NiRT/v,  ...  where    Ni,  N2...   represent 
the  number  of  mols  of  the  respective  substances;  and  the  partial 
'pressure  of  any  of  the  substances  is  evidently  equal  to  the  total  pres- 
j  sure  of  the  gas  multiplied  by  the  ratio  of  the  number  of  mols  of  that 
i  substance  to  the  total  number  of  mols  in  the  mixture.    This  ratio  is 
^commonly  called  the  mol-fraction  (x)  of  that  substance.    That  is, 
V=f  *i,  where  xi=  N,/(  #!  +  #,+  .. .). 

Air  affords  an  important  example  of  these  considerations.  Pure 
dry  air  has  at  different  times  and  places  a  fairly  constant  composition; 
namely,  it  contains  approximately  21.0  mol-percent  of  oxygen  (O2), 
78.0  mol-percent  of  nitrogen  (N2),  0.94  mol-percent  of  argon  (A),  and 
0.03  mol-percent  of  carbon  dioxide  (CO2),  and,  in  accordance  with 
Dalton's  law,  the  partial  pressures  of  these  four  substances  in  air  are 
21.0,  78.0,  0.94,  and  0.03  percent,  respectively,  of  the  total  pressure 
of  the  air.  As  will  be  shown  later,  it  is  these  partial  pressures,  not  the 
total  pressure  of  the  air,  which  determine  the  quantities  of  the  respec- 
tive substances  which  dissolve  in  water  when  it  is  shaken  with  air, 
and  it  is  the  partial  pressure  of  the  oxygen  which  determines  the 
equilibria  of  oxidation  reactions  taking  place  in  air. 

Applications  of  the  Law  of  Partial  Pressures.  — 

Prob.  8.  —  A  Bessemer  converter  is  charged  with  10000  kg.  of  iron 
containing  3%  of  carbon,  a.  How  many  cubic  meters  of  air  at  27°  and 
i  atm.  are  needed  for  the  combustion  of  all  the  carbon,  assuming  one 
third  to  burn  to  COj  and  two  thirds  to  CO?  b.  What  are  the  partial 
pressures  of  the  gases  evolved?  Ans.  a,  1950  cubic  meters. 

Prob.  9.  —  In  making  sulfur  dioxide  for  use  in  the  contact  process 
of  manufacturing  sulfur  trioxide,  30  kg.  of  sulfur  are  charged  into 
each  burner  per  hour,  and  such  an  excess  of  air  is  introduced  (to  insure 
complete  combustion)  as  to  produce  a  gas  mixture  containing  10  mol- 
percent  of  oxygen.  How  many  cubic  meters  of  air  at  20°  and  i  atm. 
must  be  introduced  per  hour?  Ans.  205  cubic  meters. 


1 6  THE  MOLAL  PROPERTIES  OF  GASES 

Determination  of  the  Dissociation  of  Gases  by  Density  Measurements.  — 
Prob.  10.  —  A  certain  fraction  7  (or  a  certain  percentage  100  7)  of 
gaseous  nitrogen  tetroxide  is  dissociated  according  to  the  reaction 
N2O4  =  2NO2  at  any  definite  pressure  p  and  temperature  T.  a.  Find  an 
expression  for  the  total  number  of  mols  i  (of  N2O4  and  NO2)  that  result 
under  these  conditions  from  one  formula-weight  M  (92  g.)  of  nitrogen 
tetroxide  taken,  b.  Find  an  expression  for  the  volume  v  that  would  be 
occupied  by  one  formula-weight  of  the  nitrogen  tetroxide  at  the  pres- 
sure p  and  temperature  T.  c.  Find  a  corresponding  expression  for  the 
density  d  of  the  vapor.  Ans.  d=Mp/(iRT). 

Prob.  ii. — Calculate  the  dissociation  of  nitrogen  tetroxide  at  50°  and 
500  mm.  from  the  following  data:  a  balloon  filled  with  it  at  this  tempera- 
ture and  pressure  weighs  71.981  g.;/the  same  balloon  when  evacuated 
weighs  71.217  g.,  and  when  filled  at  25°  with  water  (whose  density  at 
25°  is  0.997)  weighs  555.9  g.  Ans.  45%.  ^ 

13.  Determination  'of  Molecular  Weights.  —  The  (experimental 
determination  of  the  molecular  weight  of  a  gaseous  substance  consists 
in  measuring  the  volume  o!^,  weighed  quantity  of  the  substance  at  an 
observed  pressure  and  temperature.  From  these  quantities  (v,  m,  p, 
and  T)  the  molecular  weight  of  the  substance  is  calculated  with  the 
aid  of  the  perfect-gas  equation. 

Prob.  12.  —  Experimental  Determination  of  Vapor-Density  by  Hof- 
mann's  Method.  —  0.1035  g-  of  a  volatile  liquid  is  introduced  into  the 
vacuum  above  a  mercury  column  in  a  graduated  tube  standing  in  an 
open  vessel  of  mercury.  The  tube  is  entirely  surrounded  by  a  jacket 
through  which  steam  at  100°  is  passed.  The  mercury  column  falls 
till  it  stands  260  mm.  above  the  mercury-level  in  the  vessel  below,  and 
the  volume  of  the  completely  vaporized  liquid  is  observed  to  be  63.0 
ccm.  The  barometric  pressure  at  the  time  is  752  mm.  At  100°  the 
density  of  mercury  is  13.35,  and  its  vapor-pressure  is  negligible.  Make 
a  sketch  showing  the  apparatus;  and  calculate  the  density  and  molecu- 
lar weight  of  the  vapor.  Ans.  d =0.001645;  M  =  76.8. 

Prob.  13.  —  Experimental  Determination  of  Molecular  Weight  by  the 
Air-Displacement  Method  of  Victor  Meyer.  —  A  cylindrical  bulb  pro- 
vided with  a  long  stem  is  filled  with  dry  air;  and  it  is  heated,  while 
still  open  to  the  atmosphere,  to  a  constant  temperature  (of  about  218°) 
within  a  jacket  filled  with  vapor  of  boiling  naphthalene.  The  stem, 
which  protrudes  above  the  vapor  jacket,  has  a  side-arm  which  delivers 
into  a  graduated  tube  filled  with  water  and  inverted  over  water,  so 
that  the  air  to  be  later  expelled  from  the  bulb  can  be  collected  and 
measured.  A  tiny  glass  bottle  containing  0.2210  g.  of  a  volatile  liquid 
substance  is  dropped  into  the  heated  bulb,  and  at  the  same  moment 
the  bulb  is  cut  off  from  communication  with  the  atmosphere  by  closing 
the  top  of  the  stem.  The  liquid  rapidly  vaporizes  and  drives  out  of 
the  bulb  a  volume  of  air  equal  to  the  volume  occupied  by  the  vapor 


PRESSURE-VOLUME  RELATIONS  OF  GASES  17 

of  the  substance,  the  pressure  being  always  that  of  the  atmosphere. 
This  air  collected  in  the  measuring  apparatus  is  found  to  have  a  volume 
of  24.65  ccm.  at  22°  under  the  barometric  pressure  of  752  mm.;  but  of 
this  pressure  20  mm.  is  due  to  the  pressure  of  water  vapor.  Make  a 
sketch  showing  the  apparatus;  and  calculate,  a,  the  number  of  mols  of 
air  expelled,  and  b,  the  molecular  weight  of  the  vaporized  substance. 
Ans.  a,  0.000980;  b,  225. 

Since  gases  at  atmospheric  pressure  conform  to  the  perfect-gas  laws 
only  approximately,  and  since  gaseous  densities  are  not  commonly 
determined  with  so  great  accuracy  as  the  composition  of  substances, 
the  exact  value  of  the  molecular  weight  of  a  compound  is  usually 
derived  from  the  gravimetric  composition,  the  density  being  employed 
only  to  determine  what  multiple  or  submultiple  of  the  value  so  derived 
is  in  accordance  with  Avog;adro's  principle. 

Prob.  14.  —  Exact  Evaluation  of  Molecular  Weights.  —  a.  A  certain 
oxide  contains  exactly  72.73%  of  oxygen.  What  does  this  show  in 
regard  to  its  molecular  weight?  b.  At  o°  and  i  atm.,  one  liter  of  this 
(gaseous)  oxide  is  found  to  weigh  exactly  1.977  g-  What  is  the  mo- 
lecular weight  of  the  oxide  corresponding  to  this  datum?  c.  What 
is  the  exact  molecular  weight  derived  by  considering  the  data  relating 
both  to  the  composition  and  density?  d.  What  do  these  two  values 
of  the  molecular  weight  show  as  to  the  percentage  deviation  of  the 
density  from  that  which  is  required  by  the  perfect-gas  equation? 
Ans.  d,  0.68%. 

14.  Derivation  of  the  Atomic  Weights  of  Elements  and  of  the 
Molecular  Composition  of  Compounds.  —  The  exact  values  of  atomic 
weights  are  based  on  analytical  determinations  of  the  combining 
weights,  as  described  in  Art.  5.  The  multiple  of  the  combining  weight 
adopted  as  the  atomic  weight  of  any  element  is  derived  by  finding  the 
smallest  weight  of  the  element  contained  hi  one  molecular  weight  of 
any  of  its  gaseous  compounds.  This  is  the  true  atomic  weight  only  in 
case  some  one  of  the  compounds  studied  contains  in  its  molecule  a 
single  atom  of  the  element;  and  the  adopted  atomic  weight  is  therefore 
strictly  only  a  maximum  value,  of  which  the  true  atomic  weight  may 
be  a  submultiple.  The  probability  that  the  true  atomic  'weight  has 
been  found  evidently  increases  with  the  number  of  the  gaseous  com- 
pounds whose  molecular  weights  have  been  determined. 

The  multiples  of  the  combining  weights  adopted  as  the  atomic 
weights  have,  however,  not  been  derived  solely  from  molecular-weight 
determinations.  From  the  laws  relating  to  certain  other  properties, 
mentioned  in  Art.  5,  independent  values  of  the  atomic  weights  have 


1 8  THE  MOLAL  PROPERTIES  OF  GASES 

been  obtained,  which  confirm  and  extend  those  derived  from  molecular 
weights. 

The  molecular  composition  of  a  gaseous  substance  (that  is,  the 
number  and  nature  of  the  atoms  in  its  molecule)  can  evidently  be  de- 
rived from  its  molecular  weight,  its  composition  by  weight,  and  the 
atomic  weights  of  the  elements  contained  in  it.  The  chemical  formu- 
las of  substances  whose  molecular  weights  in  the  gaseous  state  are 
known  are  ordinarily  so  written  as  to  express  this  molecular  com- 
position. Such  formulas  are  called  molecular  formulas. 

Derivation  of  Atomic  Weights  and  Molecular  Formulas.  — 

Prob.  15.  —  Certain  compounds  have  molecular  weights  M  (referred  to 
that  of  oxygen  as  32)  and  percentages  of  oxygen  x  as  follows:  Sulfur  tri- 
oxide,  M  =80.07, x  =  59-95  5  water,  M  =  18.02,  x  =  88.80;  carbon  dioxide, 
M  =44.00,  x  =  72.73.  a.  Find  the  weight  of  oxygen  contained  in  one 
molecular  weight  of  each  of  these  oxides,  b.  What  conclusion  as  to 
the  atomic  weight  of  the  element  oxygen  can  be  drawn  from  these 
results  (which  are  based  on  the  convention  that  the  molecular  weight 
of  oxygen  gas  is  32)?  c.  What  does  this  show  as  to  the  number  of 
atoms  in  the  molecule  of  oxygen  gas? 

Prob.  16.  —  a.  Calculate  the  combining  weight  of  the  element  con- 
tained in  the  oxide  whose  composition  was  given  in  Prob.  140. 
b.  What  conclusion  as  to  its  atomic  weight  can  be  drawn  from  this 
combining  weight  and  the  density  given  in  Prob.  146?  c.  What  con- 
clusion can  be  drawn  as  to  the  molecular  formula  of  the  oxide? 

Pr*.  b.  17.  —  a.  The  chloride  of  a  certain  element  is  found  by  analysis 
to  contain  52.50%  of  chlorine,  whose  atomic  weight  is  35.46;  and  it  is 
found  to  have  at  150°  a  vapor-density  4.71  as  great  as  that  of  air  at  the 
same  temperature  and  pressure.  What  conclusion  can  be  drawn  from 
these  facts  as  to  the  exact  atomic  weight  of  the  element?  b.  The 
hydride  of  the  same  element  is  a  gas  which  contains  5.91%  of  hydrogen; 
and  it  is  found  to  be  produced  without  change  in  volume  when  hydrogen 
(H2),  whose  atomic  weight  is  1.008,  is  passed  over  the  solid  elementary 
substance.  What  conclusion  can  now  be  drawn  as  to  the  atomic 
weight  of  the  element?  c.  What  are  the  simplest  molecular  formulas 
of  the  hydride  and  of  the  chloride  consistent  with  these  conclusions? 

Prob.  18.  —  A  certain  hydrocarbon  is  composed  of  92.25%  of  carbon 
and  7.75%  of  hydrogen,  whose  atomic  weights  are  12.00  and  1.008, 
respectively.  Its  density  when  in  the  form  of  vapor  at  100°  and  i  atm. 
is  2.47  times  as  great  as  that  of  oxygen  under  the  same  conditions. 
Calculate  its  exact  molecular  weight,  and  derive  its  molecular  formula. 

The  molecules  of  elementary  substances  may  consist  of  single 
atoms  or  of  two  or  more  atoms.  Thus  the  molecular  formulas  of 
some  of  those  whose  density  in  the  gaseous  state  has  been  determined 
are :  H2,  N2,  O2,  F2,  C12,  Br2, 12,  P4,  As4,  He,  Ne,  A,  Hg,  Cd,  Zn,  Na,  K. 


PRESSURE-VOLUME  RELATIONS  OF  GASES  19 

At  very  high  temperatures  some  of  these  have  been  shown  to  dissociate 
into  simpler  molecules;  thus  above  1500°  the  molecule  of  iodine  con- 
sists of  a  single  atom. 

A  knowledge  of  the  molecular  formulas  of  substances  is  of  impor- 
tance principally  because  the  chemical  relationships  of  different  sub- 
stances are  far  more  clearly  brought  out  by  such  formulas  than  by 
the  simpler  ones  expressing  merely  composition  by  weight.  The 
structure  theory,  which  underlies  the  science  of  organic  chemistry, 
is  based  upon  a  knowledge  of  the  molecular  weights  of  substances. 

15.  Deviations  from  the  Perfect-Gas  Laws  at  Moderate  Pres- 
sures. —  The  deviations  from  the  perfect-gas  law,  so  long  as  they  do 
not  exceed  a  few  percent,  are  almost  exactly  proportional  to  the 
pressure.  This  fact  is  expressed  by  the  following  equation,  which 
will  be  called  the  gas  equation  at  moderate  pressures: 


In  this  equation  the  coefficient  o^eyidently  represents  the  fractional 
deviation  at  unit-pressure  of  the  actual  value  of  p  v  from  that  required 
by  the  perfect-gas  law.  The  value  of  a  varies  with  the  nature  of  the 
gas,  and  for  a  given  gas  with  the  temperature. 

The  following  table  shows  the  values  of  the  percentage  deviation 
(100  a)  of  pv  from  NRT  at  one  atmosphere  for  various  gases  at  o°, 
together  with  their  condensation-  temperatures  at  one  atmosphere. 

DEVIATIONS  AT  MODERATE  PRESSURES 

Formula  of  gas               He        H2         N2      NO       CO2  NH3  SO2 

Percentage  deviation  -f  0.06  +0.05   —0.04   —0.12   —0.68  —1.52  —2.38 

Condensation-temp.    —269°  —253°  —196°  —151°   —78°  —34°  —  10° 

Prob.  19.  —  Calculation  of  the  Deviation-Coefficient.  —  a.  Calculate 
the  value  of  the  deviation  coefficient  a  for  nitrous  oxide  (N2O)  at  o°  from 
the  fact  that  its  density  at  o°  and  i  atm.  is  0.001978  g.  per  can.  b. 
State  what  percentage  error  would  be  made  in  calculating  by  the  perfect- 
gas  equation  the  volume  of  this  gas  at  o°  and  3  atm.  Ans.  a,  —0.0073. 

16.   Pressure-Volume  Relations  of  Gases  at  High  Pressures.  —  In 

the  figure  are  plotted  the  values  of  pv/T  in  atmospheres  and  cubic 
centimeters  as  ordinates  against  the  values  of  the  pressure  in  atmos- 
pheres as  abscissas  for  one  mol  of  hydrogen  at  o°,  of  nitrogen  at  o°,  of 
carbon  dioxide  at  60°  and  at  40°,  and  of  a  perfect  gas  (marked  P.  G. 
in  the  figure).  At  temperatures  between  o°  and  60°,  helium  (He)  and 
neon  (Ne)  have  curves  similar  to  that  of  hydrogen  (H2);  oxygen  (O2), 
carbon  monoxide  (CO),  and  nitric  oxide  (NO)  have  curves  similar 


20 


THE  MOLAL  PROPERTIES  OF  GASES 


to  that  of  nitrogen;  and  nitrous  oxide  (N2O),  ammonia  (NH3),  and 
ethylene  (CzH.^)  have  curves  similar  to  those  of  carbon  dioxide  (CO2). 


20 


100  200 

Pressure  in  Atmospheres 


300 


Pressure-Volume  Relations  of  Gases  at  High  Pressures.  — 

Prob.  20.  —  Estimate  with  the  aid  of  the  figure  the  percentage 
amount  by  which  at  60  atm.  the  volume  of  H2  at  o°,  of  N2  at  o°,  and 
of  CO2  at  40°  and  at  60°  Deviates  from  that  of  a  perfect  gas. 

Prob.  21.  —  Summarize  the  general  conclusions  in  regard  to  the 
pressure-volume  relations  of  gases  that  can  be  drawn  from  the  curves 
of  the  figure,  from  the  statements  in  the  preceding  text,  and  from  the 
condensation-temperatures  given  in  Art.  15. 

Prob.  22.  —  Determine  from  the  figure  how  many  kilograms  of  CO2 
can  be  charged  into  a  1 5-liter  steel  tank  which  will  safely  wit1  stand  a 
pressure,  a,  of  50  atm.;  6,  of  100  atm.;  c,  of  150  atm.  Assume  ixiat  the 
tank  might  be  exposed  (under  a  summer  sun)  to  a  temperature  a  -  high 
as  40°.  Ans.  a,  1.7;  6,  9.6;  c,  11.5. 


PRESSURE-VOLUME  RELATIONS  OF  GASES  21 

The  pressure-  volume-  temperature  relations  of  gases  are  closely 
represented  up  to  moderately  high  pressures  by  the  following  expres- 
sions, known  as  the  van  der  Wads  equation,  in  which  a  and  b  are 
constants  (with  respect  to  changes  of  volume,  pressure,  and  tempera- 
ture) varying  with  the  nature  of  the  gas: 


(p  +  ^r)  (o-Nb)  =  NRT;  or  for  one  mol,  (p+Q  (v-b)  =  RT. 

With  reference  to  the  volume  occurring  in  the  second  of  these  ex- 
pressions, it  may  be  noted  that,  in  general,  quantities  referred  to  one 
mol  are  called  molal  quantities,  and  those  referred  to  one  gram  specific 
quantities;  and  that  in  this  book  the  former  are  designated  by  a  wavy 
line,  and  the  latter  by  a  straight  line,  above  the  symbol  of  the  quantity. 
Thus,  v  denotes  the  molal  volume,  the  volume  of  one  mol,  and  v 
denotes  the  specific  -volume,  the  volume  of  one  gram,  of  substance. 
The  term  specific  is  also  applied  to  quantities  which  are  referred  to 
unit-values  of  other  determining  factors  such  as  the  volume,  length, 
or  cross-section;  thus  specific  resistance  denotes  the  electrical  resist- 
ance of  a  mass  of  the  substance  one  centimeter  long  and  one  square 
centimeter  in  cross-section. 

The  applicability  of  this  equation  and  its  theoretical  significance  are 
discussed  in  Art.  20. 

Any  equation  which,  like  the  van  der  Waals  equation,  the  perfect- 
gas  equation,  or  the  gas  equation  at  moderate  pressures,  expresses  the 
relation  between  the  volume,  pressure,  and  temperature  of  a  sub- 
stance within  a  certain  range  of  conditions  is  called  the  equation  of 
state  of  the  substance  under  those  conditions. 


22  THE  MOLAL  PROPERTIES  OF  GASES 

II.    THE   KINETIC   THEORY 

17.  The    Fundamental    Kinetic    Hypotheses.  —  The    molecular 
theory  has,  with  the  aid  of  certain  hypotheses  in  regard  to  the  motion  of 
molecules,  been  developed  into  a  body  of  principles,  collectively  known 
as  the  kinetic  theory.  ^The  two  most  fundamental  of  these  hypotheses, 
which  express  from  a  molecular  viewpoint  the  nature  of  pressure  and 
of  temperature,  respectively,  are  as  follows: 

The  molecules  of  gaseous  or  liquid  substances  are  continually  in 
motion  in  all  directions,  the  motions  taking  place  in  accordance  with 
the  ordinary  laws  of  mechanics.  The  molecules  are  constantly  collid- 
ing with  one  another  and  with  the  walls  of  the  containing  vessel;  but 
no  loss  of  kinetic  energy  results  from  these  collisions.  The  pressure  of 
the  substance  on  the  walls  is  the  integrated  effect  of  the  impacts  of  its 
molecules.  This  may  be  called  the  kinetic  pressure  hypothesis. 

At  the  same  temperature  the  molecules  of  different  fluid  (that  is, 
gaseous  or  liquid)  substances  have  the  same  kinetic  energies,  whatever 
be  the  masses  of  their  molecules  or  the  pressure  or  volume  of  the 
substance;  that  is,  for  two  different  substances  at  the  same  tempera- 
ture, \rn\u?  =  J/W2M22,  where  m\  and  nh  represent  the  masses  of  the 
two  kinds  of  molecules,  and  u\  and  u^  their  velocities.  The  kinetic 
energy  of  the  molecules  increases  with  the  temperature,  and  it  is 
determined  solely  by  it.  Mathematically  expressed,  Jww2=f(D, 
where  i(T)  is  a  function  which  is  identical  for  all  fluid  substances. 
This  may  be  called  the  kinetic  temperature  hypothesis. 

18.  The  Kinetic  Equation  for  Perfect  Gases.  —  From  the  kinetic 
pressure  hypothesis  may  readily  be  derived  an  expression  in  terms  of 
the  molecular  quantities  for  the  pressure-volume  product  of  a  perfect 
gas,  as  shown  in  the  following  problem.    In  this  derivation  two 
simplifying  assumptions,  admissible  in  the  case  of  a  perfect  gas,  are 
made;  namely,  that  the  volume  of  the  molecules  is  negligible  in  com- 
parison with  the  volume  of  the  gas,  and  that  any  force  of  attraction 
between  the  molecules  is  negligible  because  of  the  relatively  large 
mean  distances  between  them.    This  expression  is: 


In  this  equation  n  denotes  the  number  of  molecules  in  the  volume  v 
of  the  gas  at  the  pressure  />,  m  the  mass  of  a  single  molecule,  and  u 
the  velocity  of  the  molecules.  It  may  be  called  the  kinetic  equation 
for  perfect  gases. 


THE  KINETIC  THEORY  23 

Prob.  23.  —  Derivation  of  the  Kinetic  Equation.  —  A  volume  v  of  a 
perfect  gas  is  contained  in  a  cubical  vessel  whose  edges  have  the  length 
/.  The  gas  consists  of  n  molecules,  each  of  mass  m.  Consider  that 
all  the  molecules  are  moving  with  the  same  velocity,  u,  and  that  the 
motions  of  these  molecules,  which  in  reality  take  place  in  all  directions, 
are  resolved  in  the  three  directions  perpendicular  to  the  faces  of  the 
cube,  which  is  equivalent  to  assuming  that  one  third  of  all  the  molecules 
is  moving  in  each  of  these  directions,  a.  Find  the  number  of  impacts 
which  any  one  molecule  makes  on  the  opposite  faces  in  a  unit  of  time. 
b.  Find  the  total  number  of  impacts  which  all  the  molecules  make  on 
all  the  faces  in  a  unit  of  time.  c.  Show  that  the  number  of  impacts 
which  the  molecules  make  on  a  unit  of  surface  in  a  unit  of  time  is  j^nu/v. 
d.  Find  the  change  of  momentum  which  attends  each  impact,  taking 
account  of  the  fact  that  the  molecule  rebounds  from  the  wall  with  a 
velocity  equal  to  that  with  which  it  strikes  it.  e.  Find  the  change  of 
momentum  of  all  the  molecules  that  strike  a  unit  of  surface  in  a  unit  of 
time.  /.  Noting  that  this,  in  accordance  with  the  general  equation 
of  mechanics  f=m(du/di),  is  equal  to  the  force  (/)  exerted  on  the  unit 
of  surface  or  to  the  pressure,  show  that  the  pressure-volume  product 
is  expressed  by  the  equation  pv  = 


The  derivation  given  in  the  preceding  problem  is  not  conclusive, 
since  it  involves  an  assumption  as  to  the  directions  of  motion  of  the 
molecules;  but  a  rigorous  analysis  leads  to  the  same  kinetic  equation. 
It  is  important  to  note,  however,  on  account  of  its  bearing  on  certain 
surface  phenomena,  that  the  rigorous  treatment,  which  takes  into 
account  the  fact  that  molecules  approach  the  surface  not  only  normally 
but  obliquely  in  all  directions,  leads  to  the  conclusion  that  the  actual 
number  of  impacts  made  by  all  the  molecules  on  unit-surface  in  unit- 
time  is  larger  than  that  derived  above;  this  actual  number  being 
given  by  the  expression  0.23  nu/v,  instead  of  by  ^nu/v. 

From  the  kinetic  equation  Boyle's  law  may  be  immediately  de- 
duced. For,  since  a  change  only  in  the  volume  of  the  gas  will  not 
cause  a  variation  in  the  number  of  molecules  n  (unless  they  are  thereby 
caused  to  associate  or  dissociate),  and  since  the  kinetic  energies 
%mu2  of  the  separate  molecules  will  remain  constant,  in  accordance 
with  the  kinetic  temperature  hypothesis,  so  long  as  the  temperature 
remains  unchanged,  it  follows  that  also  the  product  p  v  will  undergo 
no  change  when  the  volume  is  varied  at  a  constant  temperature. 

The  law  of  temperature-effect  is  also  a  consequence  of  the  kinetic 
equation;  for  any  definite  cnange  in  temperature  must  evidently 
cause  the  same  change  in  the  pressure-  volume  product  of  different  gases, 
since  it  causes  the  same  change  in  the  kinetic  energy  of  their  molecules. 


24  THE  MOLAL  PROPERTIES  OF  GASES 

The  principle  of  Avogadro  also  is  readily  derived  from  the  kinetic 
equation  for  perfect  gases  and  from  the  kinetic  temperature  hypothesis. 
Thus,  for  two  perfect  gases  with  equal  pressure-volume  products, 
piVi=pzV2,  and  therefore  \n\m\u^  =\n^m^\  and  for  the  case  that 
they  have  also  equal  temperatures,  \rn\u?  =^m^u^;  whence  follows 
HI  =«2.  That  is,  those  quantities  of  different  perfect  gases  which  at 
any  definite  temperature  have  equal  pressure-  volume  products  consist 
of  an  equal  number  of  molecules. 

Certain  principles  relating  to  the  velocity  of  the  molecules,  and  even 
the  exact  absolute  values  of  their  velocities,  can  be  derived  from  the 
kinetic  equation  in  the  ways  illustrated  by  the  following  problems. 

Velocity  of  the  Molecules.  — 

Prob.  24.  —  a.  Show  how  the  velocities  of  the  molecules  of  different 
gases  at  the  same  temperature  are  related  to  the  molecular  weights  of 
the  gases;  illustrating  by  hydrogen  and  oxygen.  b.  Show  how  the 
velocities  of  the  molecules  of  the  same  gas  at  different  temperatures  are 
related  to  those  temperatures;  illustrating  by  oxygen  at  o°  and  100°. 

Prob.  25.  —  Calculate  the  velocity  of  the  molecules  of  oxygen  at  20° 
in  kilometers  and  in  miles  per  second.  Ans.  0.478  km.;  0.30  mile. 

19.  The  Kinetic  Energy  of  the  Molecules  and  the  Avogadro  Num- 
ber. —  An  expression  may  be  readily  derived  for  the  kinetic  energy 
K  due  to  the  translatory  motion  of  the  n  molecules  present  in  any 
mass  of  a  perfect  gas.  Namely,  for  this  kinetic  energy,  which  is 
obviously  equal  to  \nmu^^  the  following  expression  is  obtained  at  once 
by  combining  the  kinetic  equation  for  perfect  gases,  pv  =lnmu2,  with 
the  perfect-gas  equation,  pv  =  NRT: 


From  this  equation  the  increase  dK/dT  of  the  kinetic  energy  of  the 
molecules  in  one  mol  of  gas  per  degree  of  temperature  is  seen  to  be  f  R, 
which  is  readily  shown  to  have  the  value  12.474  ergs  or  2.978  (ap- 
proximately 2.98)  calories. 

Prob.  26.  —  Kinetic  Energy  of  the  Molecules.  —  a.  Calculate  the 
translatory  kinetic  energy  in  ergs  and  calories  of  the  molecules  in  one 
mol  of  a  perfect  gas  at  20°.  b.  Show  that  the  increase  in  this  kinetic 
energy  per  degree  is  2.98  cal.  Note  that  i  cal.  equals  4.182  Xio7  ergs. 

The  corresponding  expression  for  the  kinetic  energy  of  a  single 
molecule  evidently  is: 

±        2     3  RT       3RT 

J  m  u2  =  f  —/-ft  =  f  -—  • 
2  n/  N      *  n 


THE  KINETIC  THEORY  25 

The  quantity  n/  N,  which  represents  the  number  of  molecules  in 
one  mol  of  any  substance  (for  example,  in  2.015  grams  of  hydrogen  or 
32.00  grams  of  oxygen),  is  a  fundamental  constant  known  as  the 
Awgadro  number  (n).  Its  value  has  been  recently  determined  by 
a  variety  of  independent  methods.  The  most  probable  value  is 
6.06  X1023. 

The  above  derived  expression  for  %mu2  is  the  general  form,  holding 
true  for  all  fluid  substances,  of  the  functional  relation  between  tem- 
perature and  kinetic  energy  of  the  separate  molecules,  which  is  postu- 
lated in  the  kinetic  temperature  hypothesis.  For,  since  according  to 
that  hypothesis  this  kinetic  energy  is  the  same  function  of  the  tempera- 
ture for  all  fluid  substances,  and  since  it  has  been  shown  to  be  pro- 
portional to  the  absolute  temperature  for  a  perfect  gas,  it  follows  that 
it  is  proportional  to  the  absolute  temperature  for  all  fluid  substances. 
In  other  words,  the  mathematical  expression  of  the  kinetic  tempera* 
ture  hypothesis,  which  was  given  in  the  indefinite  form  Jww2=f(T), 
may  now  be  written  in  the  definite  form  %mu2=%kT;  where  k  is 
a  constant  (equal  to  R/n)  with  respect  to  any  variation  whatsoever 
of  the  nature  or  state  of  the  fluid  substance.  The  value  of  the 
fundamental  constant  k  is  readily  calculated  from  the  previously 
given  values  of  R  and  n,  and  is  thus  found  to  be  i.372Xio~16  ergs 
per  degree. 

Prob.  27.  —  The  Number  and  the  Mass  of  the  Molecules.  —  a.  Show 
that  the  number  of  molecules  in  i  ccm.  of  a  perfect  gas  at  20°  and  i  atm. 
is  2.51  Xio19.  b.  Calculate  the  mass  of  one  molecule  of  hydrogen,  and 
that  of  one  atom  of  chlorine. 

20.  The  Kinetic  Equation  for  Imperfect  Gases.  —  As  the  pressure 
of  a  gas  increases,  the  two  effects  that  were  neglected  in  the  deriva- 
tion of  the  kinetic  equation  for  perfect  gases  (Art.  18)  must  be  taken 
into  account;  namely,  the  effect  of  the  volume  occupied  by  the  mole- 
cules themselves,  and  the  effect  of  forces  of  attraction  between  the 
molecules.  These  influences  affect  respectively  the  two  factors  in- 
volved in  the  derivation  of  the  equation  —  the  number  of  impacts  on 
the  unit  of  surface  in  the  unit  of  time,  and  the  force  corresponding  to 
each  impact.  For  the  number  of  impacts  made  by  a  row  of  molecules 
moving  between  the  opposite  faces  of  the  cube  will  evidently  be 
increased  when  the  distance  which  the  molecules  have  to  traverse 
becomes  appreciably  shortened  by  collisions  between  them.  And  the 
force  produced  by  the  impact  of  each  molecule  will  evidently  be 


26  THE  MOLAL  PROPERTIES  OF  GASES 

diminished  if,  as  it  approaches  the  wall,  it  is  appreciably  attracted 
by  the  molecules  behind  it  so  that  it  strikes  the  wall  with  a  smaller 
momentum. 

The  correction  to  be  applied  to  the  kinetic  equation  for  perfect 
gases  for  the  volume  of  the  molecules  may  be  derived  by  considering 
the  number  and  character  of  the  collisions  (whether  head-on,  glancing, 
or  tangential)  and  the  shortening  of  the  path  by  each  type  of  collision, 
and  by  integrating  the  separate  effects.  It  is  evident  that  the  resultant 
effect  will  be  dependent  on  the  size  of  the  molecules;  and  under  certain 
simplifying  assumptions  it  can  be  shown,  if  the  volume  of  one  mole- 
cule be  represented  by  </>,  that  the  distance  between  the  opposite  faces 
of  the  cube  actually  traversed  by  the  molecules  is  reduced  in  conse- 
quence of  the  collisions  from  /  to  /  (v— 4«0)/».  The  number  of  im- 
pacts by  all  the  molecules  on  unit-surface  in  unit-time  is  increased  in 
the  same  proportion,  namely, 

£        nu  .      £       j.-n.          Nj  nu 

from—  (as  found  in  Prob.  230)  to  —, r- 

6v  6(v-4wt>) 

*The  effect  of  the  attraction  of  the  molecules  on  the  momentum 
with  which  a  molecule  strikes  the  wall  can  be  derived  from  the  follow- 
ing considerations.  The  decrease  in  its  momentum  mu  must  be  pro- 
portional to  the  force  acting  on  it  and  to  the  time  during  which  that 
force  acts.  The  force  of  attraction  must,  however,  be  proportional  to 
the  number  of  molecules  n/v  in  the  unit  of  volume;  and  the  time 
during  which  the  molecule  is  subjected  to  this  attraction,  which  is  the 
time  required  for  it  to  trasverse  the  outer  layers  of  the  gas,  must 
be  inversely  proportional  to  its  velocity,  which  is  uv/(v—  4/fc/>);  this 
velocity  being  greater  than  u,  the  velocity  between  collisions,  because 
of  the  just  considered  effect  of  the  volume  of  the  molecules.  The 
change  in  momentum  attending  the  impact  of  the  molecule  on  the 
wall  will  therefore  be  reduced 

from  2mu  to  2 (mu 2  (v — 4^0) ) , 

where  a  is  a  proportionality-constant  dependent  on  the  nature  of  the 
gas  under  consideration.  The  change  of  momentum  of  all  the  mole- 
cules striking  the  unit  of  surface  in  the  unit  of  time,  which  is  equal  to 

*  Articles,  paragraphs,  and  problems  to  which  an  asterisk  is  preSxed  may  be  omitted  in 
briefer  courses. 


THE  KINETIC  THEORY  27 

the  pressure  p,  is  found  by  multiplying  this  quantity  by  the  quantity 
derived  in  the  preceding  paragraphs.    The  product  so  obtained  is 

nmuz        an2 
3(fl-4«0)     3*>2 

*It  is  evident  from  this  equation  that  the  externally  manifested 
pressure  p  is  the  resultant  of  an  outward-directed  pressure  (repre- 
sented by  the  first  term  of  the  equation)  caused  by  the  motion  of  the 
molecules,  and  an  inward-directed  pressure  (represented  by  the  second 
term)  caused  by  the  attraction  of  the  molecules  within  the  gas  for 
those  in  the  outer  or  surface  layers. 

By  considering  the  corrections  for  the  volume  and  attraction  of 
the  molecules  there  may  be  derived  (in  the  way  just  shown)  the 
following  expression,  which  may  be  called  the  kinetic  equation  for 
imperfect  gases: 


In  this  equation  <f>  denotes  the  volume  of  a  single  molecule  of  the 
substance,  and  a  represents  a  constant  characteristic  of  the  sub- 
stance which  expresses  the  effect  of  the  attraction  of  its  molecules. 
Now,  since  it  has  been  shown  (in  Art.  18)  that  for  any  gaseous 
or  liquid  substance  \nmuz  =  NRT,  and  since  the  number  of  mole- 
cules n  may  evidently  be  replaced  in  the  second  and  fourth  terms  by 
the  number  of  mols  N  of  the  substance  with  new  proportionality- 
constants,  this  expression  may  be  transformed  into  the  following  one, 
known  as  the  van  der  Waals  equation,  already  mentioned  in  Art.  16: 


By  considering  one  mol  of  the  substance,  N  becomes  equal  to  unity, 
and  the  equation  assumes  the  simpler  form  which  is  often  conveniently 
employed. 

It  is  evident  from  the  derivation  that  the  quantity  b  in  the  van  der 
Waals  equation  denotes  four  times  the  volume  of  the  molecules  in  one 
mol  of  the  gas;  and  that  the  term  aN2/vz  or  aft2  denotes  the  inward- 
directed  pressure  due  to  the  attraction  of  the  molecules.  This  pres- 
sure is  called  the  cohesion  pressure;  and  the  constant  a  may  be  called 
the  cohesion  constant. 


28  THE  MOLAL  PROPERTIES  OF  GASES 

Since  the  cohesion-constant  a  and  the  volume-constant  b  vary  with 
the  nature  of  the  molecule,  it  is  evident  that  the  van  der  Waals  equa- 
tion involves  the  assumption  that  the  molecules  of  the  substance  do 
not  undergo  a  change  through  association  as  the  pressure  increases,  or 
through  dissociation  as  the  temperature  rises.  These  effects  are,  in  fact, 
of  not  infrequent  occurrence;  and  the  limits  of  applicability  of  the  equa- 
tion must  therefore  be  determined  experimentally  for  each  substance. 

With  many  substances,  probably  with  all  of  those  whose  molecules 
are  all  of  one  kind,  the  van  der  Waals  equation  gives  values  of  the 
pressure  or  volume  which  agree  with  the  observed  values  within  one 
or  two  percent,  so  long  as  the  pressure  does  not  much  exceed  20 
atmospheres,  or  so  long  as  the  molal  volume  is  not  less  than  1000  ccm. 
But  it  may  lead  to  very  erroneous  results  when  applied,  as  it  some- 
times is,  at  much  higher  pressures  (for  example,  to  the  computation  of 
the  critical  constants).  At  these  higher  pressures  a  satisfactory 
correspondence  with  the  observed  values  can  be  attained  by  expressing 
the  quantities  a  and  b  as  definite  functions  of  the  molal  volume,  in- 
stead of  regarding  them  as  constants. 

A  comparison  of  the  observed  pressures  with  the  calculated 
pressures  given  in  the 'following  table  illustrates  the  validity  of  the 
van  der  Waals  equation  for  a  few  simple  gases  at  molal  volumes 
of  2000,  1000,  and  500  ccm.,  or  at  pressures  between  12  and  40 
atmospheres.  The  constants  used  in  calculating  the  pressures  are 
given  in  the  last  two  columns.  In  evaluating  them  the  pressure 
is  expressed  in  atmospheres  and  the  volume  in  cubic  centimeters, 
and  they  are  referred  to  one  mol  of  the  gas.  They  are  so  chosen 
as  to  reproduce  the  observed  pressure- volume  relations  accurately 
at  pressures  up  to  10  or  12  atmospheres. 

VALIDITY  OF  THE  VAN  DER  WAALS  EQUATION 


Sub- 
stance 

Temper- 
ature 

Molal 
Volume 

Observed 
Pressure 

Calculated 
Pressure 

Constants 
io"«  a 

b 

N2 

0° 

IOOO 

22.2 

22.2 

1.26 

47 

500 

44.2 

44.4 

C02 

40° 

2000 
IOOO 
500 

12.  1 
22.8 
40.6 

12.2 
23.2 
44.1 

6.23 

128 

NH3 

0° 

IOOO 

500 

I6.5 
23.1 

I6.7 
26.4 

IO.O4 

163 

H20 

200° 

IOOO 

500 

29.9 
46.8 

30.1 

57-8 

10.02 

210 

THE  KINETIC  THEORY  29 

*Prob.  28. — Applications  of  the  van  der  Waals  Equation.  —  The 
constants  a  and  b  in  the  van  der  Waals  equation  have  for  carbon  dioxide 
the  values  6.23Xio6  and  128,  respectively,  when  the  pressure  is  ex- 
pressed in  atmospheres  and  the  volume  in  cubic  centimeters,  a.  Cal- 
culate the  pressure  of  carbon  dioxide  at  40°  when  the  volume  of  one 
mol  of  the  gas  is  2000  ccm.;  b,  when  it  is  1000  ccm.  c.  Find  the  ratio 
of  these  computed  pressures  to  the  actual  pressures,  which  are  12.11 
and  22.84  atm.,  respectively,  d.  Find  the  ratio  of  the  cohesion  pres- 
sures to  these  two  external  pressures,  e.  Find  the  ratio  of  the  correc- 
tion for  the  volume  of  the  molecules  to  the  volume  of  the  gas  for  these 
two  cases.  /.  Find  the  ratio  of  the  cohesion  pressure  to  the  total 
pressure,  and  the  ratio  of  the  volume  correction  to  the  total  volume, 
for  carbon  dioxide  at  o°  and  i  atm.  (where  the  molal  volume  is  not 
greatly  different  from  22,400  ccm.,  that  for  a  perfect  gas),  g.  From 
these  ratios  derive  the  percentage  deviation  of  pi)  from  RT  for  carbon 
dioxide  at  o°  and  i  atm. ;  and  compare  the  result  with  that  given  in  the 
table  in  Art.  15.  Ans.  a,  12.16  atm.;  b,  23.24  atm. 

*Prob.  2g.  —  Determination  of  the  Constants  of  the  van  der  Waals  Equa- 
tion. —  When  one  mol  of  NH3  has  a  volume  of  2000  ccm.  at  30°  its 
pressure  is  11.03  atm->  and  when  the  gas  is  heated  at  this  (constant) 
volume  to  60°,  its  pressure  becomes  12.37  atm.  Calculate  the  values 
in  atmospheres  and  cubic  centimeters  of  the  constants  a  and  b  for  NHs 
in  the  van  der  Waals  equation.  Ans.  a  =  io.o3Xio6;  6  =  163. 

*Prob.  30.  —  Principles  Relating  to  the  Internal  Pressures.  -*-  a.  From 
the  results  of  the  preceding  problem  find  the  values  of  the  two 
oppositely  directed  internal  pressures  of  which  the  external  pressure 
of  11.03  atm.  is  the  resultant,  for  NH3  at  a  molal  volume  of  2000  ccm. 
at  30°.  b.  Find  the  values  of  the  two  internal  pressures  at  60°;  and 
state  the  principles  which  determine  the  effect  of  temperature  upon 
each  of  them  when  the  gas  is  kept  at  constant  volume,  c.  Find  the 
value  of  the  cohesion  pressure  at  30°  when  the  molal  volume  is  20,000 
ccm.  Ans.  a,  2.51  and  13.54  atm.;  c,  0.0251  atm. 

21.  The  Characteristics  of  the  Molecules.  —  From  the  van  der 
Waals  volume-constant  and  the  Avogadro  number  the  volume  and 
diameter  of  a  single  molecule  can  be  readily  calculated,  as  illustrated 
by  Prob.  31.  From  the  diameter  of  the  molecule  the  number  of 
its  collisions  per  second  and  the  mean  distance  traversed  between 
successive  collisions  can  be  derived  as  shown  in  the  text  below. 

Prob.  ji. — Determination  of  the  Dimensions  of  the  Molecules. — 
a.  From  the  values,  128  and  163  ccm.,  of  the  volume-constants  for 
CO2  and  for  NH3  calculate  the  actual  volume  of  the  molecules  in  one 
mol  of  each  of  these  gases,  and  in  1000  ccm.  of  each  of  the  gases  at  20° 
and  i  atm.  b.  Assuming  the  molecules  are  spherical,  calculate  with 
the  aid  of  the  Avogadro  number  the  diameters  of  the  molecules  of  CO2 
and  NH3  in  centimeters. 


30  THE  MOLAL  PROPERTIES  OF  GASES 

The  results  of  determinations  of  these  molecular  magnitudes  for  two 
substances,  hydrogen  and  carbon  dioxide,  at  20°  and  one  atmosphere 
are  given  in  the  following  table.  The  number  of  molecules  per  cubic 
centimeter  and  the  velocity  of  the  molecules  are  for  comparison  also 

included. 

MOLECULAR  MAGNITUDES 

Hydrogen  Carbon  dioxide 

Number  of  molecules  per  cubic  centimeter     .        2.5Xio19  2.5Xio19 

Velocity  in  centimeters  per  second      .      .      •.       i.pXio5  0.4  Xio5 

Diameter  of  the  molecule  in  centimeters        .       2.4Xio~8  3.3Xio~8 

Collisions  by  any  one  molecule  per  second      .  io.2Xio9  6.2Xio9 

Mean  distance  between  collisions  in  centimeters  i7.2Xio~6  6.iXio~6 

These  figures  show  that  at  atmospheric  pressure  the  number  of 
collisions  per  second  is  enormous  and  that  the  distance  which  on  the 
average  a  molecule  traverses  between  collisions  is  extremely  small. 
They  show,  however,  that  this  distance  is  several  hundred  times 
greater  than  the  diameter  of  the  molecule. 

*From  the  diameter  of  the  molecule,  as  stated  above,  certain  other 
quantities  characteristic  of  the  molecular  condition  of  gases  can  be 
derived.  The  most  important  of  these  are  the  number  of  collisions 
between  the  molecules  per  second,  and  the  average  distance  traversed 
by  a  molecule  between  the  collisions,  a  quantity  commonly  called  the 
mean  free  path.  An  approximate  relation  between  the  diameter  of 
the  molecules  and  these  quantities  and  a  fairly  close  estimate  of  their 
magnitudes  can  be  derived  as  illustrated  by  Probs.  32  and  33,  with  the 
aid  of  the  approximate  assumptions  that  the  molecules  are  spherical, 
that  any  one  molecule  under  consideration  is  moving  with  a  uniform 
velocity  equal  to  that  required  by  the  kinetic  equation,  and  that  the 
other  molecules  among  which  it  is  moving  are  stationary.  The 
number  of  collisions  C  per  second  which  one  molecule  of  radius  r 
moving  with  velocity  u  makes  with  the  other  molecules  when  n  of 
these  are  present  in  volume  t>,  and  the  mean  free  path  /  of  the  molecules, 
are  thus  found  to  be  given  by  the  expressions: 


*It  is  seen  that,  as  would  be  expected,  the  number  of  collisions  per 
second  is  proportional  to  the  number  of  molecules  (n/v)  per  unit 
volume,  to  the  cross-section  (irr2)  of  the  molecules,  and  to  their  velocity 
(u);  and  that  the  mean  distance  traversed  between  collisions  is  in- 


THE  KINETIC  THEORY  jx 

versely  proportional  to  the  number  of  molecules  and  to  their  cross- 
section.  By  a  more  rigorous  analysis,  not  involving  the  above  stated 
simplifying  assumptions  that  the  molecule  under  consideration  has  a 
uniform  velocity  equal  to  that  required  by  the  kinetic  equation  and 
that  the  other  molecules  are  stationary,  it  can  be  shown  that  the 
mean  free  path  (for  spherical  molecules)  is  only  0.71  times  as  great  as 
that  computed  by  the  above  equation. 

Relations-  between  the  Diameter  of  the  Molecules,  the  Number  of  Col- 
lisions, and  the  Mean  Free  Path.  — 

*Prob.  J2.  —  a.  Considering,  as  illustrated  in  the  figure,  that  a 
molecule  of  radius  r  is  moving  with  a  velocity  u  among  stationary 
molecules,  find  an  expression  for  the 
volume  of  the  space  within  which  the 
centers  of  the  stationary  molecules 
must  be  located  in  order  that  the  ""TTV  ^ 
moving  molecule  may  collide  with  — ^-^ — 
them  during  one  second,  b.  Find 
an  expression  for  the  number  of 
stationary  molecules  with  which  the  moving  molecule  collides  per 
second,  in  case  a  volume  v  of  the  gas  contains  n  molecules  uniformly 
distributed,  c.  Derive  from  this  equation  the  expression  given  in  the 
preceding  text  for  the  mean  free  path  of  the  molecules,  stating  the 
reasons  involved. 

*Prob.  jj.  —  From  the  diameters,  of  the  C02  and  NH3  molecules  calcu- 
lated in  Prob.  31,  find  for  the  case  that  each  gas  is  at  20°  and  i  atm. 
the  number  of  collisions  made  by  one  molecule  per  second,  and  the 
mean  free  path. 

*The  mean  free  path  is  a  factor  that  determines  the  magnitudes  of 
certain  properties  of  gases,  especially  of  their  viscosity  and  their  elec- 
trical and  thermal  conductivities;  and  conversely,  its  own  magnitude 
can  be  computed  from  the  results  of  measurements  of  these  properties. 
Since  the  number  of  collisions  and  the  diameter  of  the  molecules  can 
be  calculated  from  the  mean  free  path  by  means  of  the  equations  given 
above,  there  are  available  a  number  of  independent  methods  of  deter- 
mining all  these  molecular  quantities. 

22.  Distribution  of  the  Velocities  and  Kinetic  Energies  of  the 
Molecules  of  a  Gas.  —  It  has  been  assumed  thus  far  that  all  the 
molecules  of  a  gas  have  the  same  velocity  and  kinetic  energy.  Evi- 
dently this  cannot  be  the  case,  since  as  a  result  of  the  collisions  there 
is  a  constant  interchange  of  the  kinetic  energies,  with  consequent 
variation  of  their  values  from  zero  up  to  values  much  greater  than 
the  mean  kinetic  energy.  It  can  be  shown  that,  in  view  of  the  very 


32 


THE  MOLAL  PROPERTIES  OF  GASES 


large  number  of  molecules  and  the  frequency  of  their  collisions,  their 
kinetic  energies  and  velocities  must  be  distributed  in  accordance  with 
the  laws  of  probability,  of  which  the  distribution  of  errors  of  observa- 
tion around  the  mean  value  affords  a  familiar  illustration.  This  law 
of  the  distribution  of  kinetic  energies  and  velocities  is  known  as  the 
Maxwell  distribution  law. 


0.7 
0.6 
D.6I 


I  0.3 

0.21 


0.1 


E- 


K  .05     dK 


1.0  1.5 

Kinetic  Energy  K 


2.0 


2.5 


3.0 


The  figure  shows  graphically  the  distribution  of  kinetic  energies 
required  by  this  law.  The  two  graphs  OABC  and  ODEF  refer  to  o° 
and  136.5°,  corresponding  to  absolute  temperatures  which  stand  to 
each  other  in  the  ratio  2:3.  As  abscissas  are  plotted  the  kinetic 
energies  K  (equal  to  %tmi2)  of  the  molecules  expressed  in  terms  of  the 
mean  value  of  the  kinetic  energy  at  o°  as  unity;  and  as  ordinates  are 
plotted  the  values  P  of  such  a  function  of  K  that  the  product  P  dK, 
represented  by  the  area  beneath  any  element  of  the  graph,  is  equal  to 
the  fraction  dn/n  of  the  whole  number  of  molecules  n  (in  one  mol) 
that  have  kinetic  energies  between  K  and  K-\-dK\  the  area  beneath 
the  whole  graph  being  equal  to  unity  and  corresponding  to  the  total 
number  of  the  molecules.  Otherwise  interpreted,  the  product  P  dK 
is  the  fraction  of  the  time  that  any  given  molecule  has  kinetic  energies 
between  K  and  K-\-dK.  Correspondingly,  the  relative  values  of  the 
ordinates  P,  for  different  points  of  the  curve  represent  the  relative 
fractions  of  all  the  molecules  that  have  kinetic  energies  within  any 


THE  KINETIC  THEORY  33 

definite  infinitesimal  amount  of  those  represented  by  the  correspond- 
ing abscissas  K;  or  they  represent  the  relative  fractions  of  the  time 
that  any  one  molecule  has  these  kinetic  energies.  In  the  case  of  the 
graph  OA  BC,  where  the  kinetic  energies  are  expressed  in  terms  of  the 
mean  kinetic  energy  at  the  same  temperature,  the  quantity  P,  com- 
monly called  the  probability  function,  has  the  form 


Prob.  34.  —  Fraction  of  the  Molecules  having  Kinetic  Energies  between 
Definite  Limits.  —  Determine  from  the  figure  the  percentages  of  the 
total  number  of  molecules  which  at  o°  have  kinetic  energies  lying  be- 
tween limits  which  are  multiples  of  the  mean  kinetic  energy  at  o°,  as 
follows:  a,  o.o  and  0.2;  b,  0.2  and  0.6;  c,  0.6  and  i.o;  d,  i.o  and  2.0. 

Prob.  55.  —  Effect  of  Temperature  on  the  Fraction  of  the  Molecules 
having  Kinetic  Energies  Greater  than  any  Definite  Absolute  Value.  — 
Determine  from  the  figure  the  percentages  of  all  the  molecules  that, 
a,  at  o°,  and  b,  at  136.5°,  have  kinetic  energies  greater  than  twice  the 
mean  kinetic  energy  at  o°. 

The  fact,  illustrated  by  the  preceding  problem,  that  the  fraction 
of  the  molecules  having  kinetic  energies  exceeding  any  definite  value 
increases  rapidly  with  rising  temperature  is  of  much  importance  in 
relation  to  the  effect  of  temperature  on  the  vapor-pressure  of  liquid 
substances  and  on  the  rate  of  chemical  reactions. 

The  graphs  in  the  figure  represent  also  the  distribution  of  the 
squares  of  the  velocities  of  the  molecules  in  terms  of  the  mean  velocity- 
square  at  o°.  Similar  graphs  can  be  drawn  to  show  the  distribution 
of  the  velocities  themselves.  When  the  velocity  u  is  expressed  in 
terms  of  the  square-root  of  the  mean  value  of  the  velocity-square  at 
the  same  temperature,  the  probability  function  has  the  form: 


The  percentages  of  the  total  number  of  molecules  which  possess 
values  of  the  velocity  that  lie  between  limits  which  are  certain  multi- 
ples of  the  square-root  of  the  mean  velocity-square  are  as  follows: 
Velocity  multiples       .      .     0.0-0.4    0.4-1.4    1.4-2.5     2.5-00 
Percentage  of  molecules  .          8.2          76.9         12.7        2.2 

*Differential  equations  expressing  the  distribution  of  kinetic  ener- 
gies and  velocities  can  readily  be  formulated  as  in  Prob.  36.  These 
equations  cannot  be  directly  integrated;  but  by  methods  of  approxi- 
mation the  integrals  have  been  evaluated  between  various  limits. 


34  THE  MOLAL  PROPERTIES  OF  GASES 

*Prob.  36.  —  Expressions  for  the  Fraction  of  the  Molecules  having 
Definite  Kinetic  Energies  or  Definite  Velocities.  —  With  the  aid  of  the 
probability-functions  given  in  the  text,  formulate  differential  equations 
for  the  fraction  dn/n  of  the  total  number  of  molecules  n  which  possess, 

a,  kinetic  energies  (expressed  in  terms  of  the  mean  kinetic  energy) 
between  any  value  K  and  an  infinitesimally  greater  value  K+dK;  and 

b,  velocities  (expressed  in  terms  of  the  square-root  of  the  mean  velocity- 
square)  between  any  value  u  and  an  infinitesimally  greater  value  u+du. 

*Expressions  for  the  fractions  dn/n  and  n0/n  of  the  molecules  that 
at  any  temperature  T  have  absolute  kinetic  energies  (i)  between 
E  and  E-\-dE,  and  (2)  greater  than  any  definite  value  EQ,  are  given 
below.  Equation  (i)  is  obtained  from  that  of  Prob.  360,  by  replacing 
the  energy-ratio  K  by  the  quantity  JS/f  k  T  (  J  k  T  being  the  mean 
kinetic  energy  of  the  molecules);  and  equation  (2)  is  obtained  by 
integrating  equation  (i)  between  the  limits  EQ  and  oo  : 


*The  mean  kinetic  energy  at  any  definite  temperature  is  evident- 
ly directly  related  to  the  square-root  of  the  mean  velocity-square 
(\/Sw2/w)  at  the  same  temperature;  this  square-root  being  equal  to 
the  velocity  derived  from  the  kinetic  equation  %nmu2  =  NRT.  In 
many  kinetic  relations  the  most  probable  kinetic  energy  and  the  most 
probable  "velocity  are  also  important  quantities.  These  are  denned  to 
be  the  kinetic  energy  and  velocity,  respectively,  which  the  greatest 
number  of  molecules  possess.  Their  values  can  be  shown,  as  in 
Prob.  37,  to  stand  in  a  definite  numerical  ratio  to  the  mean  values 
just  mentioned.  Thus  the  square  of  the  most  probable  velocity  is 
just  two-thirds  of  the  mean  velocity-square.  Another  important  quan- 
tity is  the  mean  velocity  (2u/ri),  which  can  be  shown  by  a  less  simple 
process  to  be  0.921  of  the  square-root  of  the  mean  velocity-square. 

*Prob.  37.  —  Relations  between  the  Most  Probable  and  Mean  Values 
of  the  Kinetic  Energy  and  of  the  Velocity.  —  a.  Determine  from  the 
figure  the  ratio  between  the  mean  kinetic  energy  and  the  most  probable 
kinetic  energy,  b.  Derive  this  relation  also  from  the  probability- 
function  given  in  the  text,  noting  that  the  most  probable  kinetic  energy 
corresponds  to  the  maximum  value  of  that  function,  c.  Derive 
similarly  the  relation  between  the  square-root  of  the  mean  velocity- 
square  and  the  most  probable  velocity. 


THE  ENERGY  RELATIONS  OF  GASES  35 

III.     THE   ENERGY   RELATIONS   OF    GASES 

23.  Energy  in  General  and  the  Law  of  its  Conservation.  —  The 
essential  idea  underlying  the  concept  of  energy  is  the  constancy  of  a 
quantity  which  is  involved  in  all  the  changes  taking  place  in  the 
universe;  and  this  is  often  explicitly  expressed  by  the  statement  that 
energy  is  neither  created  nor  destroyed  in  any  process  whatever. 
This  statement  is  called  the  law  of  the  conservation  of  energy,  or  the 
first  law  of  thermodynamics. 

The  law  may  be  stated  more  concretely  as  follows:  When  a  quan- 
tity of  energy  disappears  at  any  place,  a  precisely  equal  quantity  of 
energy  simultaneously  appears  at  some  other  place  or  places;  and 
when  a  quantity  of  energy  disappears  in  any  form,  a  precisely  equal 
quantity  of  energy  simultaneously  appears  in  some  other  form  or 
forms;  equal  quantities  of  energy  of  different  forms  being  understood 
to  be  such  quantities  as  produce  the  same  effect  (for  example,  in 
modifying  motion  or  raising  temperature)  when  converted  into  the 
same  form. 

The  exactness  of  this  law  has  been  established  by  many  careful 
quantitative  investigations  made  for  the  purpose.  The  law  is  also 
confirmed  by  the  correspondence  of  the  conclusions  drawn  from  it  with 
well-established  facts  and  principles.  Among  these  may  be  mentioned 
the  following  principle,  which  is  a  conclusion  based  upon  the  failure 
of  many  attempts  to  produce  a  contrary  result:  The  production  of 
an  unlimited  amount  of  work  by  a  machine  or  arrangement  of  matter 
which  receives  no  energy  from  the  surroundings  is  an  impossibility. 
An  ideal  process  like  that  here  stated  to  be  impossible  is  sometimes 
called  perpetual  motion  of  the  first  kind  (to  distinguish  it  from  another 
kind  of  perpetual  motion  which  will  be  later  described). 

Of  especial  importance  in  chemical  considerations  are  the  energy 
effects  attending  changes  in  the  state  of  systems.  By  the  system  is 
understood  the  definite  kinds  and  quantities  of  matter  under  con- 
sideration. A  system  is  said  to  be  in  a  definite  stale  when  the  tem- 
perature, pressure,  state  of  aggregation,  quantity,  and  chemical 
composition  of  each  of  its  parts  is  fixed;  and  a  change  in  any  of  these 
conditions  is  called  a  change  in  the  state  of  the  system.  Thus  the  system 
under  consideration  might  consist  of  16  g.  of  the  element  oxygen  and 
2.015  g-  of  the  element  hydrogen.  This  system  might  exist  initially 
in  the  state  of  a  mixture  of  0.5  mol  of  oxygen  gas  (O2),  and  i  mol  of 
hydrogen  gas  (H2)  at  a  temperature  of  20°  and  a  total  pressure  of  ten 


36  THE  MOLAL  PROPERTIES  OF  GASES 

atmospheres.  It  might  now  undergo  a  change  in  state,  and  exist 
finally  in  the  state  of  one  formula-weight  of  liquid  water  at  20°  and 
one  atmosphere. 

It  follows  from  the  law  of  the  conservation  of  energy  that,  when 
the  state  of  a  system  is  fixed,  its  energy-content  U  is  also  fixed;  and 
therefore  also  that  any  change  in  the  state  of  a  system  is  attended  by 
a  definite  change  AU  in.  its  energy-content  (equal  to  the  difference 
Uz—Ui  between  its  energy-content  in  the  two  states),  whatever  be 
the  process  by  which  the  change  takes  place.  This  corollary  from 
the  law  of  the  conservation  of  energy,  stating  that  the  change  in  the 
energy-content  of  a  system  is  determined  solely  by  its  initial  and 
final  states,  is  of  so  much  importance  in  chemical  considerations  that 
it  has  received  a  special  name  —  the  law  of  initial  and  final  states. 

Corresponding  to  the  change  in  the  energy-content  of  the  system, 
there  must,  of  course,  be  an  energy  effect  in  the  surroundings;  and,  in 
general,  the  only  method  of  determining  the  change  in  the  energy- 
content  of  a  system  is  to  measure  the  energy  produced  in  the  sur- 
roundings. This  is  ordinarily  done  by  causing  the  change  in  state 
to  take  place  within  a  calorimeter  and  measuring  the  heat  which  is 
imparted  to  or  withdrawn  from  the  calorimeter  and  its  contents; 
account  being  taken  of  any  other  form  of  energy  which  may  at  the 
same  time  be  produced  or  destroyed. 

The  energy  lost  or  gained  by  the  system  may  appear  or  disappear  in 
the  surroundings  in  any  of  its  various  forms;  but  in  energy  considera- 
tions it  is  primarily  important  only  to  differentiate  the  production 
of  heat  from  that  of  the  other  forms  of  energy.  Such  other  forms  of 
energy  (that  may  be  associated  with  matter)  are  collectively  desig- 
nated work.  Under  this  term  are  included,  for  example,  production  of 
motion  in  a  body,  displacement  of  a  force  through  a  distance,  change 
of  volume  under  pressure,  development  of  electrical  energy,  and 
production  of  chemical  changes.  All  these  forms  of  work  are  quanti- 
tatively transformable  into  one  another;  but  the  transformation  of 
heat  into  work  is  subject  to  certain  limitations,  which  will  be  considered 
later.  It  is  for  this  reason  that  heat  and  work  are  differentiated  from 
each  other.  It  will  be  noted  that  the  term  work  is  here  used  in  a 
broader  sense  than  that  in  which  it  is  used  in  the  science  of  mechanics. 

With  the  aid  of  the  concepts  of  energy-content,  heat,  and  work, 
the  law  of  the  conservation  of  energy  may  now  be  expressed  by  the 
statement  that  the  decrease  in  the  energy-content  of  the  system 


THE  ENERGY  RELATIONS  OF  GASES  37 

when  it  undergoes  a  change  in  state  is  equal  to  the  sum  of  the  quanti- 
ties of  heat  and  work  produced  in  the  surroundings.  Of  these  three 
quantities  the  change  in  energy-content  is  most  fundamental,  since 
it  has  a  definite  value  for  any  definite  change  in  state;  while  the 
quantities  of  heat  and  of  work  produced  may  vary  with  the  process  by 
which  the  change  in  state  takes  place. 

Three  units  of  energy  are  commonly  used  in  scientific  work  —  the 
erg,  the  joule,  and  the  calorie.  The  erg  is  the  energy  expended  when  a 
force  of  one  dyne  is  displaced  through  one  centimeter,  or  it  is  twice  the 
energy  possessed  by  a  mass  of  one  gram  when  moving  with  a  velocity 
of  one  centimeter  per  second.  The  joule  is  a  decimal  multiple  of  the 
erg;  namely,  one  joule  equals  107  ergs.  The  mean  calorie  is  one- 
hundredth  part  of  the  heat  required  to  raise  one  gram  of  water  from 
o°  to  100°.  This  is  identical  within  0.02%  with  the  ordinary  calorie, 
which  is  the  heat  required  to  raise  one  gram  of  water  from  15°  to  16°. 
One  calorie  (i  cal.)  is  equal  to  4.182  (approximately  4.18)  joules,  or 
4.i82Xio7  ergs,  this  value  being  the  so-called  mechanical  equivalent 
of  heat. 

Prob.  38.  —  Evaluation  of  the  Change  in  Energy-Content.  —  When  i  g. 
of  liquid  water  is  vaporized  at  100°  against. a  pressure  of  i  atm.,  the 
heat  withdrawn  from  the  surroundings  is  537  cal.,  and  the  work  pro- 
duced by  the  expansion  against  the  constant  pressure  is  168  joules. 

a.  In  which  state  does  the  water  have  the  greater  energy-content,  and 
what  is  the  difference  between  the  two  energy-contents  in  calories? 

b.  How  much  heat  would  be  drawn  from  the  surroundings  in  case  i  g. 
of  liquid  water  at  100°  and  i  atm.  was  vaporized  within  a  closed  vessel 
(no  work  being  produced  in  the  surroundings)  so  as  to  form  water-vapor 
at  100°  and  i  atm? 

24.  Work  Attending  Volume  Changes  in  General.  —  The  work 
produced  when  a  system  changes  its  volume  is  of  especial  importance 
in  chemical  considerations.  It  is 'most  readily  derived  for  the  case 
that  the  volume  undergoes  a  change  in  dimensions  in  one  direction 
only.  Suppose  that  a  liquid  or  gaseous  substance  contained  in  a 
cylinder  is  enclosed  by  a  movable  piston  of  cross-section  a,  and  that 
a  force  /  is  exerted  upon  this  piston,  for  example,  by  a  weight  placed 
upon  it.  Suppose  now  that  the  piston  rises  through  a  distance  dl.  The 
increase  of  volume  dv  is  then  adl,  and  the  force  acting  upon  the  unit 
of  surface,  which  force  is  the  pressure  p,  is  f/a.  The  work  dW  pro- 
duced by  the  expansion  is  therefore  given  by  the  equations: 


38  THE  MOLAL  PROPERTIES  OF  GASES 

That  is,  the  work  is  equal  to  the  product  of  the  pressure  into  the 
infinitesimal  increase  of  volume  that  takes  place.  It  can  be  easily 
demonstrated  that  this  equation  also  holds  true  in  the  general  case  in 
which  the  volume  increases  in  dimensions  in  any  number  of  directions. 
The  general  expression  for  the  work  produced  when  a  body  under- 
goes a  change  of  volume  from  vi  to  %,  is  therefore: 


W 


=  I     pdo. 

*/    vi 


In  order  to  carry  out  the  integration  it  is  evidently  necessary  to  know 
how  the  pressure  varies  with  the  volume  of  the  system  as  the  change  in 
state  under  consideration  takes  place. 

The  most  common  case  in  energy  discussions  is  that  in  which  the 
external  pressure  is  considered  to  be  substantially  equal  to  the  pressure 
exerted  by  the  substance  or  substances  of  which  the  system  is  com- 
posed, these  two  pressures  differing  by  only  the  infinitesimal  amount 
which  in  the  absence  of  friction  suffices  to  cause  the  change  in  state 
to  take  place.  In  this  case,  in  order  to  carry  out  the  integration,  it  is 
necessary  to  know  the  functional  relation  between  the  pressure  and  the 
volume  of  the  substance  or  substances  involved,  —  in  other  words, 
to  know  the  equation  of  state  of  the  substance  or  mixture  of  sub- 
stances of  which  the  system  consists. 

Prob.  39.  —  Work  Produced  by  the  Change  in  Volume  of  a  System  at 
Constant  Pressure.  —  a.  Calculate  the  work  in  ergs  and  calories  that  is 
produced  when  one  formula- weight  of  water  (iH2O)  vaporizes  at  100° 
against  the  pressure  of  i  atm.  The  specific  volume  of  liquid  water  at 
100°  is  1.043,  a°d  that  of  saturated  water  vapor  is  1650.  b.  Find  the 
percentage  errors  that  would  be  made  by  neglecting  the  volume  of  the 
liquid,  and  by  regarding  the  vapor  as  a  perfect  gas.  c.  Formulate 
a  general  expression  for  the  work  produced  when  a  system  undergoes 
a  change  of  volume  under  a  constant  pressure;  and  show  that  under 
these  approximate  assumptions  the  work  W  produced  by  the  vaporiza- 
tion at  a  constant  temperature  T  of  that  quantity  of  a  liquid  which 
forms  N  mols  of  saturated  vapor  is  given  by  the  expression  W  =NRT. 
d.  Show  that  the  value  of  the  gas-constant  R  is  1.985  cal.  per  degree. 

Prob.  40.  —  Work  Producible  by  the  Change  in  Volume  and  Pressure 
of  a  Perfect  Gas  at  Constant  Temperature.  —  a.  Calculate  the  work  in 
calories  produced  when  i  mol  of  a  perfect  gas  at  2  atm.  expands  at  20° 
till  its  pressure  becomes  i  atm.,  the  opposing  external  pressure  being 
kept  always  substantially  equal  to  that  of  the  gas.  b.  Calculate  the 
work  in  calories  produced  when  under  these  conditions  iCO2  at  2  atm. 
expands  at  20°  till  its  pressure  becomes  i  atm.  The  molal  volume  of 
CO2  at  20°  and  2  atm.  is  11,890  ccm.  (or  1.16%  less  than  that  of  a 
perfect  gas).  Ans.  b,  408  cal. 


THE  ENERGY  RELATIONS  OF  GASES  39 

25.  The  Energy-Content  of  Systems  in  General  in  Relation  to 
Temperature  and  Heat-Capacity.  —  The  change  in  the  energy-content 
which  attends  a  change  in  the  temperature  of  any  system  is  experi- 
mentally determined  by  measuring  the  quantities  of  heat  and  work 
withdrawn  from  or  imparted  to  the  surroundings.  The  ratio  of  the 
quantity  of  heat  dQ  absorbed  when  the  temperature  of  the  system 
rises  from  T  to  T+dT  to  the  rise  of  temperature  dT  is  called  its  heat- 
capacity  C  at  T\  that  is,  C  =dQ/dT.  The  heat-capacity  is  substan- 
tially equal  to  the  quantity  of  heat  absorbed  when  the  temperature 
rises  one  degree.  When  the  heating  takes  place  without  change  of 
pressure,  the  heat  absorbed  per  degree  is  called  the  heat-capacity  at 
constant  pressure  Cp.  When  the  heating  takes  place  without  change 
of  volume,  the  heat  absorbed  per  degree  is  called  the  heat-capacity  at 
constant  volume  Cv.  The  heat-capacity  of  any  homogeneous  system  is 
obviously  the  product  of  its  weight  by  the  heat-capacity  of  one  gram 
of  it,  which  is  called  its  specific  heat-capacity  C  (or  often  its  specific 
heat).  The  heat-capacity  of  one  mol  or  one  formula- weight  of  a  pure 
substance  is  called  its  molal  or  formal  heat-capacity  C.  The  heat- 
capacity  of  one  atomic  weight  is  called  the  atomic  heat-capacity. 

Prob.  41.  —  Measurement  of  the  Heat-Capacity  of  Gases.  —  Through 
a  coil  within  a  water  calorimeter  at  20.000°  having  a  total  heat-capacity 
of  1 100  cal.  per  degree  n.o  g.  of  carbon  dioxide  preheated  to  various 
temperatures  were  passed  at  a  pressure  of  i  atm.  in  a  series  of  experi- 
ments, and  the  rise  in  temperature  in  the  calorimeter  was  measured, 
with  the  following  results: 

Initial  gas  temperature     .     .         200°        400°        600°        800° 
Final  calorimeter  temperature    20.389°   20.861°   21.378°   21.930° 

a.  Calculate  the  quantity  of  heat  given  off  to  the  calorimeter  in  each 
experiment,  b.  Plot  on  a  large  scale  these  heat  quantities  Q  as  ordi- 
nates  against  the  initial  gas  temperatures  t  as  abscissas,  c.  Determine 
with  aid  of  the  plot  the  molal  heat-capacity  of  carbon  dioxide  at  200° 
and  that  at  600°. 

When  a  system  is  heated  without  change  of  volume,  no  work  is 
produced,  and  the  change  in  its  energy-content  per  degree  is  therefore 
simply  equal  to  the  heat  absorbed  from  the  surroundings  or  to  the 
heat-capacity  of  the  system  at  constant  volume. 

When  a  system  is  heated  at  constant  pressure,  not  only  is  heat 
absorbed  from  the  surroundings,  but  also  work  is  produced  in  them. 
The  increase  in  energy-content  per  degree  is  therefore  equal  to  the 
heat  absorbed  diminished  by  the  work  produced. 


40  THE  MOLAL  PROPERTIES  OF  GASES 

*Prob.  42.  —  Mathematical  Expressions  for  the  Heat-Capacities.  — 
Formulate  partial-derivative  expressions  for  the  definitions  of  the  heat- 
capacity  at  constant  volume  and  that  at  constant  pressure,  and.  for 
the  relation  of  each  of  these  heat-capacities  to  the  corresponding  change 
in  energy-content  per  degree. 

Ans.  C'°=\'T\  '  e^c" 

26.  The  Energy-Content  and  Heat-Capacity  of  Perfect  Gases.  — 
Experiments  have  shown  that  when  a  perfect  gas  expands  at  a  constant 
temperature  without  producing  any  work  (for  example,  when  it 
expands  within  a  calorimeter  from  one  vessel  into  another  vessel 
previously  evacuated),  there  is  no  heat-effect  in  the  surroundings  (no 
change  of  temperature  in  the  calorimeter).  Such  experiments  have 
established  the  important  law  that  the  energy-content  of  a  definite 
quantity  of  a  perfect  gas  at  any  definite  temperature  has  the  same 
value,  whatever  be  its  volume  and  pressure.  In  other  words,  for  the 
change  in  state  which  occurs  when  a  perfect  gas  changes  its  volume 
and  pressure  at  a  constant  temperature,  the  change  At/  in  its  energy- 
content  is  zero. 

From  this  law  of  perfect  gases  the  following  principles  can  be  derived 
with  aid  of  the  law  of  initial  and  final  states,  as  in  Probs.  43  and  44 : 

(1)  At  any  definite  temperature  the  increase  per  degree  dU/dT  of 
the  energy-content  of  any  definite  quantity  of  a  perfect  gas  has  the 
same  value  whatever  be  the  initial  and  final  pressures  and  volumes; 
and  therefore  the  heat-capacity  at  constant  volume  of  any  perfect 
gas  is  independent  of  the  pressure  and  volume. 

(2)  The  molal  heat-capacity  of  a  perfect  gas  at  constant  pressure  is 
greater  than  that  at  constant  volume  by  an  amount  equal  in  value 
to  the  gas-constant  R,  whatever  be  the  gas  and  whatever  be  the 
temperature  or  pressure;  that  is, 

Cp  —  Cv  =  R  =  i  .985  cal.  per  degree. 

Derivation  of  the  Heat-Capacity  Relations  of  Perfect  Gases.  — 

Prob.  43.  —  Derive  the  first  principle  stated  in  the  preceding  text  by 

considering  that  a  perfect  gas  changes  by  two  different  processes  from  a 

volume  Vi  at  pressure  pi  and  temperature  T\  to  a  volume  %  at  pressure 

fa  and  temperature  Tz. 

Prob.  44.  —  Derive  the  second  principle  stated  in  the  preceding  text 

by  a  consideration  similar  to  that  employed  in  the  last  problem. 

This  second  principle  obviously  enables  the  heat-capacity  at  con- 
stant volume  to  be  calculated  from  the  heat-capacity  at  constant 


THE  ENERGY  RELATIONS  OF  GASES  41 

pressure.  This  is  of  importance,  since  the  former  quantity  is  one 
which  it  is  difficult  to  determine  accurately  by  direct  experiment, 
owing  to  the  fact  that  any  rigid  container  used  for  holding  the  gas 
and  preventing  its  expansion  has  unavoidably  a  much  larger  heat- 
capacity  than  the  gas  itself;  while  the  latter  quantity  can  be  readily 
and  accurately  measured  (as  illustrated  by  Prob.  41). 

In  view  of  this  difficulty  of  determining  directly  the  heat-capacity 
at  constant  volume,  it  is  important  to  note  also  that  the  heat-capacity 
ratio  Cp/Cv  can  be  derived  from  measurements  of  certain  properties  of 
gases;  especially,  from  measurements  of  the  change  of  pressure  which 
results  when  the  gas  expands  adiabatically  (that  is,  without  imparting 
heat  to  the  surroundings  or  withdrawing  heat  from  them),  and  from 
measurements  of  the  velocity  of  sound  through  the  gas.  These 
methods  need  not,  however,  be  here  described. 

27.  The  Heat-Capacity  of  Perfect  Gases  in  Relation  to  their 
Molecular  Composition.  —  The  heat-capacity  of  perfect  gases  depends 
primarily  on  the  complexity  of  their  molecules,  as  is  shown  by  the 
following  principles. 

The  molal  heat-capacity  at  constant  volume  Cv  has  the  smallest 
value  for  gases  with  monatomic  molecules,  such  as  mercury,  helium, 
and  argon;  and  it  has  the  same  value,  namely  f  R  or  2.98  cal.  for  all 
such  gases  at  all  temperatures.  The  corresponding  value  of  Cp  is  f  R 
or  4.96  cal.  It  will  be  shown  in  Art.  28  that  the  kinetic  theory  affords 
a  striking  explanation  of  this  simple  behavior. 

The  behavior  of  gases  with  diatomic  molecules  is  less  simple.  Many 
of  these  gases,  for  example  those  with  the  formulas  N2,  O2,  CO,  NO, 
HC1,  have  substantially  the  same  value  of  Cv,  and  one  which  varies 
appreciably,  but  not  very  greatly,  with  the  temperature,  between  o° 
and  1200°;  thus  for  all  these  gases  the  value  of  Cv  is  4.9  cal.  at  20°, 
5.6  cal.  at  1000°,  and  6.3  cal.  at  2000°.  Between  the  temperatures 
o°  and  -i  80°  the  values  for  these  gases  change  irregularly,  but  not 
very  greatly;  thus  at  — 180°,  Cv  is  4.73  for  nitrogen,  4.91  for  oxygen, 
and  4.76  for  carbon  monoxide.  The  values  for  hydrogen  (H2)  between 
20°  and  2000°  are  slightly  (about  0.15  cal.)  smaller  than  those  for  the 
gases  just  mentioned,  and  they  become  very  much  smaller  at  very 
low  temperatures,  attaining  at  — 180°  the  value  4.3  cal.,  and  at  —210° 
the  value  3.0  cal.  shown  by  monatomic  gases,  this  value  then  remaining 
constant  at  any  rate  down  to  -240°  (33°  A.).  A  few  diatomic  gases, 
for  example  those  with  the  formulas  C12,  Br2,  I2,  IC1,  have  at  room 


42  THE  MOLAL  PROPERTIES  OF  GASES 

temperature  larger  values  of  Cv  than  do  the  other  diatomic  gases ;  thus 
the  value  for  chlorine  is  5.9  cal.  at  20°. 

The  only  general  statements  that  can  be  made  in  regard  to  the  heat- 
capacities  of  triatomic  and  other  polyatomic  gases  are  that  the  values 
are  much  larger  than  those  for  the  diatomic  gases,  that  they  increase 
more  rapidly  with  the  temperature,  and  that  they  increase  with  the 
complexity  of  the  molecule;  thus  the  values  of  Cv  at  200°  are  6.5  for 
water  vapor  (H2O),  8.5  for  carbon  dioxide  (CO2)  and  sulfur  dioxide 
(SO2),  7.4  for  ammonia  (NH3),  and  41.6  for  ether  (C4HioO). 

The  following  table  shows  the  functions  which  express,  in 
general  within  one  or  two  percent,  the  observed  values  of  the 
molal  heat-capacities  at  constant  volume  at  atmospheric  pressure, 
of  some  of  the  common  gases  at  various  temperatures. 

MOLAL  HEAT-CAPACITIES  AT  CONSTANT  VOLUME 

Substances  Range  of  temperature  Value  of  Cv 

Hg,  Na,  K}  Complete  2.98  +0.00  T 

4.70+0.0007  r 

4. 55+0. 0007  T 
C12  0—2000°  5.65+0.00077 

NH3  0-500°  5-83  +  0.0026  T  +  0.000001 2  T2 

These  principles  in  regard  to  heat-capacity,  especially  that  relating 
to  monatomic  gases,  have  made  it  possible  to  determine  (as  illustrated 
in  Prob.  45)  the  atomic  weight  of  the  inert  elements,  helium,  neon, 
argon,  etc.,  to  which,  because  of  their  failure  to  form  compounds,  the 
usual  method  described  in  Art.  14  is  not  applicable. 

Prob.  45.  — Determination  of  Atomic  Weight  from  the  Heat-Capacity 
Ratio.  —  From  the  experimentally  determined  velocity  of  sound  in 
argon  gas  the  heat-capacity  ratio  Cp/Cv  has  been  found  to  be  1.67. 
Find  the  atomic  weight  of  argon  by  combining  this  result  with  its 
density,  which  at  20°  and  i  atm.  has  been  found  to  be  1.246  times  as 
great  as  that  of  oxygen;  stating  the  principles  involved. 

28.  The  Energy-Content  and  Heat-Capacity  of  Perfect  Gases  in 
Relation  to  the  Kinetic  Theory.  —  In  the  discussion  of  the  kinetic 
theory  in  Art.  18  only  the  translatory  kinetic  energy  of  the  molecules 
(that  due  to  their  linear  motion)  was  considered.  It  is  evident, 
however,  that  the  molecules  may  be  set  in  rotation  as  a  result  of  their 
collisions,  and  consequently  that  they  may  possess  rotational  kinetic 


THE  ENERGY  RELATIONS  OF  GASES  43 

energy.  It  is  evident  also  that  the  atoms  within  the  molecules  may 
possess  kinetic  energy  due  to  vibration  and  potential  energy  arising 
from  forces  acting  between  the  atoms  and  dependent  upon  their 
position  with  reference  to  one  another;  these  constituting  the  atomic 
energy  of  the  molecules.  The  proportion  which  each  of  these  three 
kinds  of  energy  contributes  to  the  total  energy  of  the  molecules  may 
be  expected  to  vary  with  the  temperature  and  with  the  nature  of  the 
molecules. 

In  the  case  of  monatomic  molecules  the  conditions  are  simpler. 
For  in  the  first  place,  since  the  molecule  is  identical  with  the  atom, 
the  atomic  energy  falls  away.  In  the  second  place,  it  may  well  be  true 
that  the  atoms,  and  hence  also  the  molecules,  are  so  small  or  so  sym- 
metrical that  their  kinetic  energy  of  rotation  is  negligible  in  comparison 
with  their  kinetic  energy  of  translation.  In  this  case  the  energy- 
content  of  the  gas  would  consist  solely  in  the  translatory  kinetic 
energy  K  of  its  molecules.  And  the  heat-capacity  Cv  at  constant 
volume,  which  was  shown  in  Art.  25  to  be  equal  in  general  to  the 
increase  dU/dT  in  the  energy-content  per  degree,  would  be  equal 
to  the  increase  dK/dT  in  this  kinetic  energy  per  degree;  which  increase 
dK/dT  has  been  shown  in  Art.  18  to  be  equal  to  %R  or  2.98  cal.  per 
degree  for  one  mol  of  any  gas.  Now,  experimental  determinations 
have  in  fact  shown,  as  stated  in  Art.  27,  that  all  monatomic  gases  have 
a  molal  heat-capacity  at  constant  volume  substantially  equal  to  2.98 
cal.  per  degree.  The  conformity  of  this  prediction  of  the  kinetic 
theory  with  the  facts  is  one  of  the  striking  successes  of  the  theory. 

The  fact  stated  in  Art.  27  that  most  of  the  diatomic  gases  (nitrogen, 
oxygen,  carbon  monoxide,  etc.)  have  values  of  Cv  which  are  nearly 
the  same  (about  5.0  cal.  at  20°)  for  the  different  gases  and  which  do 
not  vary  much  with  the  temperature  indicates  that  the  atomic  energies 
and  rotational  energy  of  their  molecules  increase  in  nearly  the  same 
proportion  as  the  translatory  kinetic  energy.  The  fact,  however, 
that  the  heat-capacity  of  hydrogen  gas,  though  not  differing  greatly 
from  that  of  the  other  gases  at  room  temperature,  attains  the  value 
for  a  monatomic  gas  at  very  low  temperatures,  suggests  that  its 
molecules  are  then  no  longer  taking  up  an  appreciable  proportion 
of  atomic  or  rotational  energy. 

Diatomic  gases  like  chlorine  which  have  larger  values  of  Cv  util- 
ize this  greater  amount  of  absorbed  heat  to  set  their  atoms  into  more 
rapid  vibration  and  to  separate  them  further  from  each  other.  This 


44  THE  MOLAL  PROPERTIES  OF  GASES 

is  in  correspondence  with  the  lower  temperatures  at  which  the  mole- 
cules of  these  latter  gases  dissociate  into  their  atoms. 

The  molecules  of  gases  with  more  than  two  atoms  may  be  expected 
to  take  up  more  energy  of  rotation  than  the  more  nearly  two-dimen- 
sional diatomic  molecules  and  to  convert  more  of  the  absorbed  heat 
into  atomic  energy.  This  is  in  accordance  with  the  fact  that  in 
general  the  heat-capacity  increases  with  the  complexity  of  the  molecule. 

29.  The  Energy-Content  of  Imperfect  Gases  in  Relation  to  Volume 
and  Pressure.  —  Actual  gases  show  deviations  from  the  perfect-gas 
law  that  the  energy-content  of  the  gas  at  any  definite  temperature  is 
independent  of  its  volume  and  pressure.  These  deviations  always 
lie  in  the  direction  that  the  energy-content  of  the  gas  increases  with 
decrease  in  its  pressure.  They  are  small  at  atmospheric  pressure, 
increase  rapidly  with  the  pressure,  and  become  large  at  30-1  oo  atmo- 
spheres. 

The  increase  in  energy-content  attending  an  isothermal  expansion 
is  most  accurately  determined  by  porous-plug  experiments,  in  which 
the  gas  is  caused  to  flow  continuously  through  a  well-insulated  tube 
containing  a  plug  of  porous  material,  as  illustrated  in  the  figure. 


K///////////////////////// 

///// 

////////////////////////////////////////A 

—  »|  TlPl^ 

:$$£ 
x-r-'X 

T*P**i      -> 

Under  these  conditions  the  gas  expands  without  taking  up  heat  from 
the  surroundings  and  without  producing  work,  except  that  equivalent 
to  the  change  in  its  pressure-volume  product.  It  undergoes  thereby 
a  decrease  in  temperature,  which  is  often  called  from  its  discoverers 
the  Joule-Thomson  effect.  This  decrease  in  temperature  is  accurately 
measured;  and  from  its  magnitude,  from  the  heat-capacity  of  the  gas, 
and  from  the  change  in  its  pressure-volume  product,  the  increase  in 
the  energy-content  that  would  have  attended  an  isothermal  expansion 
is  calculated,  in  the  way  illustrated  by  the  following  problem. 

*Prob.  46.  —  Determination  of  the  Change  in  Energy-Content  by  Porous- 
Plug  Experiments.  —  Carbon  dioxide  at  pressure  p\  (e.g.,  2  atm.)  and 
temperature  7\  (e.g.,  20.00°)  is  caused  to  flow  continuously  through  a 
well-insulated  hardwood  tube  containing  a  porous  plug  of  cotton.  On 
passing  through  the  plug  its  pressure  falls  to  pz  (e.g.,  i  atm.),  and  it 
emerges  from  the  tube  at  this  pressure.  After  the  gas  has  flowed  so 


ENERGY  RELATIONS  OF  GASES  45 

long  that  every  part  of  the  apparatus  has  assumed  the  temperature  of 
the  gas  in  contact  with  it,  the  expansion  of  the  gas  takes  place  without  ex- 
change of  heat  with  the  surroundings.  Its  temperature  after  passing 
through  the  plug  is  found  to  be  T2  (e.g.,  18.86°).  These  conditions  are 
illustrated  by  the  figure,  a.  What  other  process  must  be  combined 
with  this  adiabatic  process  in  order  that  the  net  result  of  the  two  pro- 
cesses may  be  the  isothermal  expansion  of  one  mol  of  the  gas  from 
pressure  pi  to  pressure  pz  at  a  temperature  7\?  b.  Formulate  expres- 
sions for  the  work  produced  W,  the  heat  absorbed  Q,  and  the  change  in 
energy-content  A £7,  for  each  of  these  two  processes.  (Note  that  in 
the  first  process  a  volume  Vi  of  the  gas  disappears  on  one  side  of  the 
plug  under  a  constant  pressure  pi,  and  that  a  certain  volume  %'  of  the 
gas  is  produced  on  the  other  side  of  the  plug  under  a  constant  pressure 
pz.)  c.  Combine  these  results  so  as  to  give  an  expression  for  the  change 
in  energy-content  that  attends  a  change  in  pressure  from  pi  to  pz  of  one 
mol  of  the  gas  at  a  constant  temperature  I\.  d.  Calculate  in  calories 
the  change  in  energy-content  attending  the  expansion  of  i  mol  of  carbon 
dioxide  from  a  pressure  of  2  atm.  to  a  pressure  of  i  atm.  at  20°,  using 
the  following  data  in  addition  to  those  given  above:  the  molal  volume 
of  carbon  dioxide  at  20°  and  2  atm.  is  11,890  can.,  and  at  20°  and  i  atm. 
is  23,920  ccm.;  its  molal  heat-capacity  at  20°  and  at  a  constant  pressure 
of  i  atm.  is  8.92  cal.  per  degree,  e.  Calculate  the  heat  that  would  be 
absorbed  from  the  surroundings  if  the  change  in  state  given  in  d  took 
place  at  20°  against  an  opposing  pressure  kept  substantially  equal  to 
that  of  the  gas,  using  the  result  obtained  in  Prob.  406.  Ans.  d,  6.78  cal. 

At  high  pressures  and  at  low  temperatures  the  Joule-Thomson 
cooling  effect  and  the  corresponding  increase  in  energy-content  attend- 
ing isothermal  expansion  are  large  even  in  the  case  of  difficultly  con- 
densible  gases.  Thus,  when  air  at  2  atmospheres  and  at  20°  expands 
to  a  pressure  of  one  atmosphere  under  the  conditions  of  a  porous-plug 
experiment,  there  is  a  fall  of  temperature  of  0.25°;  but  if  the  initial 
pressure  is  102  atmospheres  there  is  a  fall  of  25°  when  the  initial 
temperature  is  o°,  and  a  fall  of  temperature  of  74°  when  the  initial 
temperature  is  — 100°. 

Certain  technical  processes  for  liquefying  air  (for  example,  the 
Linde  process)  utilize  only  the  Joule-Thomson  effect,  but  certain 
other  processes  (such  as  the  Claude  process)  provide  for  production 
of  work  during  the  expansion,  whereby  a  greater  cooling  effect  is 
obtained  (as  indicated  by  Prob.  46  e). 

*30.  The  Energy-Content  of  Imperfect  Gases  at  Constant  Tem- 
perature in  Relation  to  the  Kinetic  Theory.  —  The  increase  in  energy- 
content  attending  the  isothermal  expansion  of  a  gas,  as  computed  from 
porous-plug  experiments  like  those  described  in  Art.  29,  is  evidently 


46  THE  MOLAL  PROPERTIES  OF  GASES 

equivalent  to  the  energy  that  must  be  imparted  to  the  gas  in  order  to 
separate  its  molecules  from  one  another.  In  other  words,  it  is  equal 
to  the  internal  work  that  has  to  be  done  in  displacing  the  cohesion 
pressure  through  the  increment  of  volume.  It  follows  therefore  that 
the  increase  dU  in  energy-content  attending  an  increase  in  volume  dv, 
and  the  increase  A  £7  in  energy-content  attending  an  increase  in 
volume  from  Vi  to  %,  will  be  given,  if  the  cohesion  pressure  has  the 
value  required  by  the  van  der  Waals  equation,  by  the  following 
expressions: 


---    ;  or  (for  one  mol 

fc      %/  \Vi 

Prob.  47.  —  Calculation  of  the  Change  in  Energy-Content  from  the 
Cohesion-Constant.  —  Calculate  the  increase  in  energy-content  in 
calories  attending  the  isothermal  expansion  of  iC02  from  2  atm.  to 
i  atm.  at  20°,  from  the  value  of  the  cohesion-constant  for  carbon 
dioxide  given  in  Prob.  28,  and  tabulate  this  value  beside  that  derived 
in  Prob.  46  from  the  data  of  porous-plug  experiments. 

Conversely,  the  cohesion-constant  can  be  derived  from  the  results 
of  porous-plug  experiments.  And  in  general  the  so-obtained  values 
of  this  constant  are  approximately  identical  with  those  calculated 
from  the  pressure-  volume  relations  by  the  van  der  Waals  equation. 
Thus  the  cohesion-constant  for  carbon  dioxide  may  be  calculated 
from  the  data  of  Prob.  46  to  be  6.62  Xio6,  while  that  derived  from  the 
pressure-volume  relations  is  6.23Xio6.  This  agreement  shows  that 
the  term  aN2/v2  in  the  van  der  Waals  equation  does  in  reality  cor- 
respond to  a  cohesion  pressure  arising  from  an  attraction  between  the 
molecules. 


CHAPTER  III 

THE  MOLAL  PROPERTIES  OF   SOLUTIONS  AND  THE 
MOLECULAR  THEORY 


I.    VAPOR-PRESSURE  AND  BOILING-POINT  IN  GENERAL 

31.  Vapor-Pressure.  —  A  liquid  in  contact  with  a  vacuous  space 
vaporizes  until  the  pressure  of  its  vapor  in  that  space  attains  a  per- 
fectly definite  value  which  is  determined  by  the  nature  of  the  liquid 
and  by  the  temperature.  If,  on  the  other  hand,  vapor  having  a 
pressure  greater  than  this  definite  value  is  brought  into  contact  with 
the  liquid,  condensation  occurs  until  the  pressure  of  the  vapor  falls 
to  that  value.  In  other  words,  for  a  given  liquid  at  a  given  tempera- 
ture there  is  only  one  pressure  which  its  vapor  can  have  and  exist 
in  equilibrium  with  that  liquid.  This  pressure  is  called  the  vapor- 
pressure  of  the  liquid.  This  is  to  be  distinguished  from  the  pressure 
of  the  vapor,  which  when  not  in  contact  with  the  liquid  may  have  any 
value  from  zero  up  to  one  somewhat  exceeding  the  vapor-pressure. 
Solids  likewise  have  definite  vapor-pressures,  which  with  certain 
substances  (like  iodine)  are  appreciable  even  at  room  temperature. 

The  vapor-pressure  of  a  liquid  or  solid  substance  increases  rapidly 
with  increasing  temperature,  as  illustrated  by  the  data  of  Prob.  3. 

When  a  liquid  or  solid  is  in  contact  with  a  space  containing  a  gas 
(for  example,  when  water  or  iodine  is  in  contact  with  an  air  space), 
approximately  the  same  quantity  of  the  liquid  or  solid  vaporizes  as 
if  the  gas  were  not  present,  provided  the  gas  is  only  slightly  soluble  in 
the  liquid,  and  provided  its  pressure  is  not  much  greater  than  one 
atmosphere.  When  the  gas  is  readily  soluble  in  the  liquid,  or  when 
its  pressure  is  large,  considerable  deviations  from  this  principle  may 
result.  * 

Prob.  i.  —  Pressure-Volume  Relations  of  Wet  and  Dry  Gases.  —  25  ccm. 
of  dry  air  at  28°  and  i  atm.  are  collected  over  water,  whose  vapor- 
pressure  at  28°  is  28  mm.  a.  What  is  the  pressure  if  the  volume  is  still 
25  ccm,?  b.  What  is  the  volume  if  the  pressure  is  i  atm.?  Ans.  b,  25.95. 

Prob.  2. — Air-Bubbling  Method  of  Determining  Vapor-Pressure. — 
2000  ccm.  of  dry  air  at  15°  and  760  mm.  are  bubbled  through  bulbs  con- 
taining a  known  weight  of  carbon  bisulfide  (CS2)  at  15°,  and  the  mix- 
ture of  air  and  bisulfide  vapor  is  allowed  to  escape  into  the  air  at  a 

47 


48  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

pressure  of  760  mm.  By  reweighing  the  bulbs,  3.01 1  g.  of  the  bisulfide 
are  found  to  have  vaporized.  Find  the  vapor-pressure  of  carbon 
bisulfide  at  15°.  Ans.  242  mm. 
Steam-Distillation  of  Liquids  Insoluble  in  Water.  — 
Prob.  j.  —  Steam  is  bubbled  through  chlorbenzene  (C6H5(?1)  in  a  dis- 
tilling flask;  and  the  vapors,  which  escape  under  a  barometric  pressure 
of  i  atm.,  are  condensed  as  a  distillate.  The  steam  partially  condenses 
in  the  distilling  flask,  and  brings  the  mixture  of  water  and  chlorbenzene 
(which  are  not  appreciably  soluble  in  one  another)  to  that  tempera- 
ture where  equilibrium  prevails  between  each  liquid  and  its  vapor. 
Determine  this  temperature  and  the  molal  composition  of  the  distillate 
with  the  aid  of  a  plot  of  the  following  data,  which  represent  the  vapor- 
pressures  of  the  pure  substances  at  various  temperatures: 

70°    80°    90°    100° 

Water    .....     234         355          526         760  mm. 
Chlorbenzene   ...       98          145          208          292  mm. 
Prob.4.  — A  current  of  steam  is  passed  at  atmospheric  pressure  through 
a  -mixture  of  water  and  nitrobenzene  (C6H6NO2).     Calculate,  a,  the 
temperature  of  the  distilling  mixture,  and,  b,  the  percentage  by  weight 
of  nitrobenzene  in  the  distillate,  from  the  following  data:  the  vapor- 
pressure  of  water  at  100°  is  760  mm.  and  changes  by  3.58%  per  degree; 
that  of  nitrobenzene  at  100°  is  20.9  mm.  and  changes  by  5.0%  per 
degree.    Ans.  b,  15.5%. 


32.  Relation  of  Boiling-Point  to  Vapor-Pressure.  —  The 

point  of  a  liquid  is  the  temperature  at  which  it  is  in  equilibrium  with 
its  vapor  when  both  are  subjected  to  any  definite  external  pressure. 
In  other  words,  it  is  the  temperature  at  which  the  vapor-pressure, 
which  increases  as  the  temperature  rises,  becomes  equal  to  the  external 
pressure.  When  this  temperature  is  exceeded  by  an  infinitesimal 
amount,  assuming  that  there  is  no  superheating,  the  vapor  forms 
throughout  the  mass  of  the  liquid  (not  merely  at  its  free  surface), 
giving  rise  to  the  familiar  phenomenon  of  boiling. 

Prob. 5.  —  Change  of  Boiling-Point  with  Barometric  Pressure.  —  The 
vapor-pressure  of  water  at  100°  increases  27.2  mm.  per  degree.  What 
variation  of  its  boiling-point  corresponds  to  a  variation  of  the  baro- 
metric pressure  from  730  to  790  mm.? 

33.  Change  of  Vapor-Pressure  with  Temperature.     The  Clapeyron 
Equation.  —  The  vapor-pressure  of  liquids   or  solids  increases  very 
rapidly  with  the  temperature,  and  roughly  by  equal  multiples  of  its 
value  for  equal  increments  of  temperature. 

Prob.  6. — Approximate  Relation  b'effryn  Vapor-Pressure  and  Tem- 
perature. —  From  the  data  of  Prob.  3  find  and  tabulate  the  ratios  of  the 


VAPOR-PRESSURE  AND  BOILING-POINT  49 

vapor-pressure  at  80°  and  70°,  90°  and  80°,  and  100°  and  90°,  a,  for 
water;  and  b,  for  chlorbenzene. 

From  the  laws  of  thermodynamics,  there  can  be  derived,  as  shown 
in  Art.  165,  an  exact  differential  expression  for  the  increase  of  the 
vapor-pressure  p  of  a  liquid  or  solid  at  the  absolute  temperature  T 
with  the  temperature.  This  expression,  which  is  known  as  the 
Clapeyron  equation,  is  as  follows: 

i£=  A# 
dT  ~  TAv 

In  this  equation  Av  represents  the  increase  in  volume  when  any  definite 
quantity  of  the  substance  changes  at  the  pressure  p  and  temperature  T 
from  the  state  of  liquid  or  solid  to  the  state  of  saturated  vapor,  and 
AH  denotes  the  heat  absorbed  from  the  surroundings  when  this 
quantity  of  substance  undergoes  the  same  change  in  state.  This 
quantity  of  heat  is  commonly  called  the  heat  of  vaporization,  or  when 
referred  to  one  mol  of  the  substance,  its  molal  heat  of  vaporization. 

Since  at  the  boiling-point  of  a  liquid  its  vapor-pressure  is  equal  to 
the  external  pressure  upon  the  liquid  and  vapor,  the  Clapeyron  equa- 
tion also  expresses  (more  clearly  in  the  inverted  form)  the  change  of 
boiling-point  with  the  external  pressure. 

In  numerical  applications  of  this  equation,  the  energy  quantities 
AH  and  Av  X  dp  must  be  expressed  in  corresponding  units.  The  latter 
quantity  will  be  in  ergs  when  the  volumes  are  in  cubic  centimeters 
and  the  pressure  is  in  dynes  per  square  centimeter.  The  units  of 
energy  commonly  employed  and  the  relations  between  them  were 
described  in  Art.  23. 

Prob.  7. — Application  of  the  Clapeyron  Equation.  —  a.  Calculate 
with  the  aid  of  the  Clapeyron  equation  the  volume  of  one  mol  of  satu- 
rated water- vapor  at  100°  from  the  following  data:  At  100°  the  vapor- 
pressure  of  water  increases  27.2  mm.  per  degree,  the  heat  of  vaporiza- 
tion of  one  gram  of  it  is  537  cal.,  and  the  specific  volume  of  liquid  water 
is  1.043.  b.  Calculate  by  the  perfect-gas  equation  the  volume  of  one 
mol  of  saturated  water-vapor  at  100°.  c.  By  comparing  these  two  values 
of  the  molal  volume,  determine  the  percentage  error  that  would  be 
made  in  assuming  that  the  saturated  vapor  conforms  to  the  perfect- 
gas  law.  Ans.  a,  29920  can.;  c,  2.3%. 

A  simpler,  but  less  exact  form  of  the  Clapeyron  equation  can  be 
derived  (as  in  Prob.  8)  by  making  the  assumptions  that  the  volume 
of  the  liquid  or  solid  is  negligible  in  comparison  with  that  of  the 


50  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

saturated  vapor  and  that  the  vapor  conforms  to  the  perfect-gas  laws. 
This  approximate  Clapeyron  equation  is 

dlogp  =  A£T 
dT     ~  RT2' 

In  this  equation  p  denotes  the  vapor-pressure  of  the  liquid  or  solid 
substance  at  the  absolute  temperature  T,  AH  its  molal  heat  of  vapori- 
zation at  that  temperature,  and  the  symbol  log  the  natural  logarithm 
(to  the  base  e).  (This  symbol  is  always  used  in  this  sense  in  this 
book;  logarithms  to  the  base  10  being  denoted  by  the  symbol  logio. 
It  may  be  noted  that  log x  =  2.3026  logio  x = approximately  2.3  logio  x.) 

The  approximate  Clapeyron  equation  evidently  yields,  at  moderate 
pressures  where  the  volume  of  the  liquid  may  usually  be  neglected, 
values  of  dp/dT  or  of  d  log  p/dT  which  are  less  than  those  derived  from 
the  exact  equation  by  nearly  the  same  percentage  amount  as  the 
actual  volume  of  the  saturated  vapor  is  less  than  that  calculated  by 
the  perfect-gas  equation,  or  by  the  same  percentage  amount  as  the 
actual  value  of  p  v  for  one  mol  of  the  saturated  vapor  is  less  than  that 
of  RT.  This  amount,  for  example,  is  2.3%  for  water  at  100°  (as 
found  in  Prob.  7),  and  3.6%  for  ethyl  alcohol  at  78.3°  (its  boiling- 
point  at  one  atmosphere).  These  quantities  are  evidently  the  values 
of  the  deviation-coefficient  a  for  these  vapors  in  the  equation  of  state 
for  gases  at  moderate  pressures,  given  in  Art.  15. 

The  approximate  Clapeyron  equation  can  be  integrated,  usually 
without  much  error,  between  any  two  temperatures  not  greatly 
different  from  each  other  under  the  assumption  that  the  heat  of 
vaporization  is  constant  between  those  temperatures,  as  illustrated 
in  Prob.  9. 

Prob.   8. — Derivation  of  the  Approximate   Clapeyron  Equation. — 

a.  From  the  exact  Clapeyron  equation  derive  the  approximate  form  given 
in  the  text  with  the  aid  of  the  simplifying  assumptions  there  mentioned. 

b.  From  the  results  of  Prob.  7  find  the  percentage  error  in  dp/dT  that 
would  result  from  neglecting  the  volume  of  the  liquid  in  the  case  of 
water  at  100°.    c.  Show  that,  aside  from  the  correction  for  the  volume 
of  the  liquid,  the  percentage  error  in  dp/dT  as  calculated  by  the  approxi- 
mate equation  is  equal  to  the  percentage  difference  between  RT  and 
the  actual  value  of  pv  for  one  mol  of  the  saturated  vapor. 

Prob.  g. — Integration  of  the  Approximate  Clapeyron  Equation. — 
a.  Integrate  the  approximate  Clapeyron  equation  so  as  to  obtain  a  rela- 
tion between  the  vapor-pressures  pi  and  p2  at  two  different  temperatures 
T\  and  T2,  assuming  that  the  heat  of  vaporization  does  not  vary  between 


VAPOR-PRESSURE  AND  BOILING-POINT  51 

the  two  temperatures,  b.  Calculate  by  the  equation  so  obtained  the 
boiling-point  of  water  at  92.0  mm.,  and  its  vapor-pressure  at  75°. 
c.  The  actual  boiling-point  at  92.0  mm.  is  49.88°,  and  the  actual  vapor- 
pressure  at  75°  is  289.3  mm-  State  what  inexact  assumptions  are 
involved  in  the  equation  which  would  account  for  the  divergence. 

34.  Vaporization  in  Relation  to  the  Kinetic  Theory.  —  When  a 
liquid  is  placed  in  communication  with  an  empty  space,  its  molecules, 
which  according  to  the  kinetic  theory  are  continually  in  motion,  will 
escape  from  its  surface  into  the  space  above  it  till  the  vapor  attains 
such  a  pressure  that  the  number  of  molecules  entering  the  liquid 
surface  from  the  vapor  side  becomes  equal  to  the  number  of  molecules 
leaving  the  surface  from  the  liquid  side.  When  this  occurs  a  stationary 
condition  of  equilibrium  is  evidently  established,  and  the  then  pre- 
vailing pressure  of  the  vapor  represents  the  vapor-pressure  of  the 
liquid. 

The  kinetic  principles  underlying  vaporization  may  be  more  fully 
developed  by  considering  what  determines  the  number  of  molecules 
that  pass  through  the  surface  in  the  two  directions. 
In  the  figure  the  line  AB  represents  an  imaginary 
plane  above  which  the  vapor  V  may  be  regarded  as 
homogeneous,  and  the  line  CD  represents  another 
imaginary  plane  below  which  the  liquid  L  may  be  A 
regarded  as  homogeneous.  In  the  intervening  region  c 
A  BCD,  constituting  the  surface  layer  of  the  liquid, 
which  in  reality  is  of  molecular  dimensions,  there 
is  a  progressive  increase  in  the  number  of  molecules 
per  unit- volume  from  the  number  present  in  the  vapor  to  the  number 
present  in  the  liquid. 

Now  the  molecules  in  the  vapor,  assuming  it  to  be  a  perfect  gas, 
exert  no  attraction  upon  one  another;  but  between  the  molecules  of  the 
liquid,  as  between  those  of  a  gas  at  high  pressure  (Art.  20),  there  is  a 
large  attraction.  Within  the  main  body  of  the  liquid  (below  CD) 
there  is,  to  be  sure,  no  resultant  attraction  on  any  given  molecule, 
since  it  is  attracted  equally  in  all  directions  by  the  uniformly  distrib- 
uted molecules  surrounding  it;  but  any  molecule  in  the  surface  layer 
will  evidently  be  attracted  downwards  (into  the  liquid)  in  consequence 
of  the  greater  density  of  the  molecules  in  the  layers  below  it.  All 
the  molecules  that  strike  the  plane  AB  from  above  (that  is,  from  the 
vapor  side)  will  therefore  have  their  downward  velocities  increased 


52  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

by  the  attraction  and  will  pass  into  the  liquid.  Of  the  molecules 
that  strike  the  plane  CD  from  below  (that  is,  from  the  liquid  side) 
only  those  will  traverse  the  surface  layer  and  pass  through  the  plane 
AB  into  the  vapor  which  have  an  upward  velocity  great  enough  to 
overcome  the  downward  attraction  exerted  by  the  molecules  within 
the  surface  layer.  In  other  words,  to  escape  from  the  liquid  a  mole- 
cule reaching  the  plane  CD  must  have  a  velocity  component  x  in  the 
upward  direction  greater  than  a  certain  limiting  value  x\,  which 
value  is  equal  to  the  decrease  in  upward  velocity  which  any  molecule 
would  experience,  in  consequence  of  the  molecular  attraction,  in 
passing  from  CD  to  AB.  The  fraction  of  the  molecules  that  have  an 
upward  velocity  component  greater  than  this  limiting  value  is  given 
by  the  Maxwell  distribution  law  (Art.  22);  and  from  this  fraction  the 
number  of  such  molecules  reaching  unit-area  of  the  plane  CD  in  unit 
time  may  be  calculated,  as  shown  in  Prob.  10.  The  number  of  mole- 
cules that  must  be  present  in  unit-volume  of  the  vapor  in  order  that 
there  may  impinge  on  unit-area  of  the  plane  AB  in  unit  time  a  number 
of  the  molecules  equal  to  that  which  escapes  into  the  vapor  is 
readily  derived  from  the  fundamental  kinetic  pressure  hypothesis, 
as  has  already  been  indicated  hi  Art.  18.  It  is  evidently  this  number 
that  determines  the  vapor-pressure  of  the  liquid  at  any  temperature. 

The  great  effect  of  temperature  on  vapor-pressure  is  readily  seen 
to  be  a  consequence  of  these  kinetic  considerations.  For  it  was 
shown  in  Art.  22  that  the  number  of  molecules  which  possess  kinetic 
energies  or  velocities  greater  than  any  definite  (relatively  large) 
kinetic  energy  or  velocity  increases  very  rapidly  with  the  temperature; 
thus  this  number  at  the  two  temperatures  is  represented  in  the  figure 
of  Art.  22  by  the  area  lying  beneath  the  two  graphs,  respectively, 
and  to  the  right  of  an  ordinate  drawn  at  the  definite  value  of  the 
kinetic  energy  (for  example,  at  the  value  K  =  2.o). 

The  heat  absorbed  by  the  vaporization  of  a  liquid  at  constant 
temperature  is  also  accounted  for  by  these  considerations.  For, 
when  any  molecule  (of  mass  m)  passes  from  the  liquid  to  the  vapor,  it 
evidently  loses  a  quantity  of  kinetic  energy  equal  to  \  m  x?\  and  cor- 
respondingly, when  one  mol  of  liquid  (containing  n  molecules)  vapor- 
izes, the  system  loses  a  quantity  of  kinetic  energy  equal  to^nm xi2, 
consequently  falls  in  temperature,  and  must  absorb  from  the  sur- 
roundings an  equal  quantity  of  heat  (or  other  energy)  if  it  is  to  return 
to  its  initial  temperature.  That  is,  the  increase  in  energy-content 


VAPOR-PRESSURE  AND  BOILING-POINT  53 

A  U  attending  the  vaporization  of  one  mol  of  liquid  at  constant 
temperature,  is  given  by  the  expression 

A£7  =  \nrnx?. 

A  striking  result  of  these  kinetic  considerations  is  that  they  lead  (as 
shown  at  the  end  of  Prob.  10)  to  the  Clapeyron  equation,  which  was 
derived,  entirely  independently,  from  the  laws  of  thermodynamics. 

*Prob.  10.  —  Kinetic  Expression  for  the  Vapor-Pressure  of  a  Liquid.  — 
a.  It  can  be  shown  with  the  aid  of  Maxwell's  distribution  law  that,  of  all 
the  molecules  n  contained  in  one  mol,  the  fraction  dn/n  which  have 
velocity-components  along  any  one  of  the  three  rectangular  axes  with 
values  between  x  and  x  +  dx  is  given  by  the  expression 
dn  0.69  _&  , 

y-'Tr*  •*-*• 

in  which  u  denotes  the  square-root  of  the  mean  velocity-square.  For- 
mulate an  expression  for  the  number  (dn/v^  of  such  molecules  in 
unit-volume  of  the  liquid,  representing  its  molal  volume  by  VL.  b.  Derive 
the  following  expression  for  the  number  dnL  of  such  molecules  that  strike 
unit-area  of  the  plane  CD  in  unit-time,  noting  that  this  number  is 
equal  to  the  number  of  them  present  in  unit-volume  multiplied  by 
their  upward  velocity-component  x  and  by  a  factor  which  represents 
the  increase  in  the  number  of  impacts  on  the  plane  resulting  (as 
described  in  Art.  20)  from  the  effect  of  the  volume  of  the  molecules, 
this  factor  being  only  a  function  of  the  molal  volume  of  the  liquid: 

n      x    ~—9 
dnL  =  0.69  77^  -  e   ™*  dx. 


c.  By  integrating  this  expression  between  the  limits  Xi  and  oo  ,  find  an 
expression  for  the  total  number  of  molecules  nL  striking  the  plane  CD 
with  a  velocity-component  greater  than  the  critical  one  Xi  (which 
barely  permits  their  passage  through  the  surface  layer),  d.  By  com- 
bining this  result  with  the  expression  nG=o.2$  un/vG  (given  in  Art.  18) 
for  the  number  of  molecules  nG  that  strike  in  unit-time  a  unit-area  of  a 
plane  bounding  a  perfect  gas  having  one  mol  (and  therefore  n  molecules) 
in  the  volume  VG,  derive  the  following  expression: 


e.  Derive  for  the  vapor-pressure  p  the  following  expression,  by  re- 
placing VG  by  the  expression  for  it  given  by  the  perfect-gas  equation: 

RT     - 


/.By  multiplying  both  numerator  and  denominator  of  the  exponent 
of  e  by  nm,  and  noting  that,  as  shown  in  the  above  text,  At/  =  \nrnx,?, 
and  that,  by  Art.  19,  \nmuz  =RT,  derive  the  expression: 


54  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

g.  Derive  the  approximate  Clapeyron  equation  by  differentiating 
the  preceding  equation  with  respect  to  T  (regarding  VL  as  constant), 
dividing  the  resulting  differential  equation  by  the  original  one,  and 
transforming;  noting  that  the  heat  A#  absorbed  by  the  vaporization  of 
one  mol  of  liquid  at  constant  pressure  and  temperature  is  equal  to  the 
increase  ACT"  in  the  energy-content  of  the  system  plus  the  work  RT 
produced  in  the  surroundings,  that  is,  that  AH=AU+RT. 


SOLUTIONS  IN  GENERAL  55 

II.    SOLUTIONS  IN  GENERAL 

35.  The  Nature  and  Composition  of  Solutions.  —  A  solution  is  a 
physically  homogeneous  mixture  of  two  or  more  chemical  substances; 
that  is,  one  which  has  no  larger  aggregates  than  the  molecules  them- 
selves. Solutions  thus  denned  may  be  gaseous,  liquid,  or  solid;  but 
only  liquid  solutions  will  be  here  considered.  When  one  substance  is 
present  in  large  proportion  it  is  called  the  solvent,  and  any  substance 
present  in  small  proportion  is  called  a  solute. 

In  considering  the  equilibrium  of  solutions  with  the  vapor  or  with 
the  solid  solvent,  the  term  phase  is  conveniently  employed.  The 
phases  of  a  system  are  its  physically  homogeneous  parts,  separated 
from  one  another  by  physical  boundaries.  Thus  any  gaseous  mixture 
or  any  solution  or  any  solid  substance  forms  a  single  phase.  A  system 
may  consist  of  any  number  of  such  phases.  Thus  a  solution  in  contact 
with  its  vapor,  or  with  the  solid  solvent,  or  with  the  solid  solute,  is 
an  example  of  a  two-phase  system.  A  solution  in  contact  both  with 
the  vapor  and  the  solid  solvent  is  a  three-phase  system. 

The  composition  of  solutions  is  often  expressed  in  terms  of  the  mol- 
fractions  of  the  substances  defined  as  in  Art.  12.  Thus,  representing, 
as  will  be  done  throughout  this  chapter,  quantities  referring  to  the 
solvent  by  letters  with  the  subscript  zero,  and  those  referring  to  the 
solute  or  solution  by  letters  without  subscripts,  the  mol-fr actions  x 
and  XQ  of  the  solute  and  solvent  in  a  solution  of  two  substances  are 
defined  by  the  equations  x  =  N/(N0+N)  and  x0=  N0/(N0+  A7), 
respectively.  The  proportion  of  solute  may  also  be  expressed  in 
terms  of  the  mol-ratio  N/  No.  Composition  is  also  expressed  in  terms 
of  the  number  of  formula-weights,  of  mols,  or  of  equivalents,  of  solute 
in  1000  grams  of  solvent.  Composition  so  expressed  will  be  desig- 
nated the  formality  f,  the  molality  c,  or  the  weight-normality  c,  respec- 
tively, of  the  solute  in  the  solution.  The  term  formality  will  be  re- 
ferred to  the  solute  as  a  whole;  but  the  terms  molality  and  weight- 
normality  will  be  used  with  reference  to  some  definite  chemical  sub- 
stance. Thus,  a  solution  containing  o.i  formula- weight  of  H2SO3  in 
1000  g.  of  water  is  o.i  formal  (o.i  f.)  in  H2SO3  (considered  without 
reference  to  the  chemical  substances  which  it  forms  in  solution) ;  but, 
owing  to  partial  ionization,  this  solution  is  0.066  molal  (0.066  m.)  or 
0.132  weight-normal  (0.132  wn.)  in  H2SO3,  and  0.034  molal  or  0.034 
weight-normal  in  H+  or  in  HSO3~.  Composition  will  in  this  book  be 
expressed  in  the  terms  of  mol-fraction,  mol-ratio,  formality,  molality, 


56  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

or  weight-normality,  unless  otherwise  stated.  Composition  expressed 
in  these  ways  has  the  advantage  of  being  independent  of  temperature. 

It  is  sometimes  convenient,  however,  to  employ  true  concentrations, 
which  term  strictly  denotes  the  quantity  of  substance  per  unit-volume. 
The  number  of  mols  of  solute  per  liter  of  solvent  (  N/VQ)  is  called  the 
mold  concentration;  and  the  solution  is  said  to  be  x  volume-molal  in 
the  solute.  Thus  a  solution  containing  o.i  mol  of  NH3  in  one  liter  of 
chloroform  at  25°  is  o.i  volume-molal  (o.i  vm.)  in  NH3  at  25°;  while 
it  is  0.0675  (weight)  molal  (0.0675  m-)  m  NH3,  since  1000  g.  of  chloro- 
form have  a  volume  of  675  ccm.  at  25°.  In  the  case  of  aqueous  solu- 
tions the  molality  and  molal  concentration  evidently  differ  from  each 
other  in  the  same  proportion  that  the  density  of  water  differs  from 
unity;  for  example,  by  0.3  percent  at  25°  and  4.3  percent  at  100°.  In 
connection  with  certain  properties,  such  as  the  electrical  conductance 
of  solutions,  which  are  directly  related  to  the  volume  of  the  solution 
(not  of  the  solvent),  there  will  be  employed  the  form  of  concentration 
familiar  in  volumetric  analysis,  called  normal  concentration,  which  is 
denned  to  be  the  number  of  equivalents  of  solute  per  liter  of  solution 
(N/v).  Thus  an  aqueous  solution  containing  i  equivalent  of  sulfuric 
acid  in  one  liter  of  solution  at  25°  is  i  normal  (i  n.)  hi  H2SC>4  at  25°; 
whereas  this  solution  may  be  calculated  with  the  aid  of  density  data 
to  be  1.020  weight-normal  in  H2SO4  at  any  temperature. 

With  reference  to  the  proportions  in  which  the  substances  are 
present,  two  groups  of  solutions  may  be  distinguished:  dilute  solutions, 
those  in  which  the  mol-fraction  of  the  solute  is  small  (not  greater  than 
c.oi  or  0.02);  and  concentrated  solutions,  those  in  which  each  sub- 
stance is  present  in  considerable  proportion.  There  is,  of  course,  no 
sharp  line  of  demarcation  between  these  two  groups  of  solutions. 

Some  types  of  concentrated  solutions  and  all  dilute  solutions  con- 
form approximately  —  more  closely  as  the  mol-fraction  of  the  solute 
approaches  zero  —  to  certain  laws,  which,  in  analogy  with  the  laws  of 
perfect  gases,  may  be  called  the  laws  of  perfect  solutions.  The  funda- 
mental laws  relating  to  the  physical  properties  of  perfect  solutions  are 
the  vapor-pressure  laws  of  Raoult  and  Henry,  the  corresponding  laws 
of  distribution  between  liquid  phases,  and  tb|(  laws  of  the  osmotic 
pressure  of  solutions.  To  consideration  of  thei&  laws  this  chapter  is 
mainly  devoted. 


VAPOR-PRESSURE  AND  BOILING-POINT  57 

III.    VAPOR-PRESSURE  AND   BOILING-POINT   OF   PERFECT   SOLUTIONS 
WITH  ONE   VOLATILE  COMPONENT 

36.  Raoult's  Law  of  Vapor-Pressure  Lowering.  —  Raoult's  law, 
referred  to  above  as  one  of  the  fundamental  laws  of  perfect  solutions, 
states  that  the  addition  o£  aAsolvent  causes  at  an^lemperature  a 
fractional  lowering  of  the  vapor-pressure  of  the  Solvent  equal  to  the 
mol-fraction  (x)  of  the  solute.  That  is: 


__    _ 
pQ      ~~NQ+N~ 

where  pQ  is  the  vapor-pressure  of  the  pure  solvent  and  p  is  its  vapor- 
pressure  in  a  solution  consisting  of  N  mols  of  solute  and  No  mols 
of  solvent.  In  this  expression  NQ  is  equal  to  the  weight  mQ  of  the 
solvent  divided  by  its  molecular  weight  Mo  in  the  vapor,  and  N  is 
equal  to  the  weight  m  of  the  solute  divided  by  its  molecular  weight 
M  in  the  solution.  The  value  of  MQ  is  ordinarily  that  corresponding 
to  the-  molecular  formula  of  the  solvent;  for,  as  stated  in  Art.  14,  the 
molecular  formula  is  commonly  so  written  as  to  represent  the  molecular 
weight  of  the  substance  in  the  state  of  a  perfect  gas. 

In  the  case  of  very  dilute  solutions  the  mol-fraction  N/(No+N) 
is  evidently  equal  to  the  mol-ratio  N/NQ;  ami  even  in  the  case  of 
moderately  dilute  solutions  the  mol-fraction  may  be  replaced  by  the 
mol-ratio  without  causing  appreciable  error. 

Although  Raoult's  law  is  exact  only  in  the  case  of  perfect  solutions, 
it  holds  true  approximately  for  all  dilute  solutions  (usually  within 
one  or  two  percent  up  to  a  mol-fraction  of  0.02),  and  for  some  con- 
centrated solutions,  as  will  be  described  in  Art.  41. 

The  application  of  Raoult's  law  to  solutions  of  moderate  concentra- 
tion affords  an  important  method  of  determining  the  approximate 
molecular  weight  and  the  molecular  composition  of  dissolved  sub- 
stances, as  illustrated  by  the  following  problems. 

Determination  of  Molecular  Weights  and  Molecular  Composition.  — 

Prob.  ii.  —  At  30°  the  vapor-pressure  of  ethyl  alcohol  (C2H5OH)  is 

78.0  mm.,  and  that  of  an  alcohol  solution  containing  5%  of  a  non- 

volatile substance  is  75.0  mm.    What  is  the  molecular  weight  of  the 

substance? 

Prob.  12.  The  experiment  described  in  Prob.  2,  ^Lrt.  31,  was  repeated, 
using  in  place  of  pure  carbon  bisulfide  atf  8.00%  solution  of  sulfur  in 
carbon  bisulfide.  2.902  g.  of  carbon  bisulfide  were  found  to  have 
vaporized.  Calculate  the  molecular  weigtit  oi^the  sulfur,  and  find  its 
molecular  formula. 


58  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

Raoult's  law  may  also  be  stated  in  the  following  simple  form,  which 
indicates  more  clearly  its  real  significance:  the  vapor-pressure  (p)  of 
the  solvent  in  a  perfect  solution  is  proportional  to  its  mol-fraction 
(#o)  ;  that  is,  representing  by  pQ  the  vapor-pressure  of  the  pure  solvent, 


Prob.  13.  —  Mathematical  Equivalence  of  the  Two  Forms  of  Raoult's 
Law.  —  a.  Show  that  the  two  statements  of  Raoult's  law  are  mathe- 
matically equivalent,  b.  Show  that  the  second  statement  of  the  law 
requires  that  the  proportionality-factor  be  the  vapor-pressure  of  the 
pure  solvent,  as  is  assumed  in  the  mathematical  expression  of  it. 

Raoult's  law  relates  fundamentally  to  the  distribution  between  the 
liquid  phase  and  vapor  phase  of  the  chemical  substance  whose  partial 
pressure  in  the  vapor  is  under  consideration.  In  other  words,  from 
the  molecular  standpoint,  it  relates  to  the  distribution  of  the  kind 
of  molecules  which  give  rise  to  this  partial  pressure.  It  shows  that 
the  number  of  these  molecules  which  are  present  in  unit-volume  of  the 
vapor  when  equilibrium  has  been  reached  is  proportional  to  the  ratio  in 
the  liquid  of  the  number  of  this  kind  of  molecule  to  the  total  number 
of  molecules  of  all  kinds.  Moreover,  since  this  molecule-ratio  is  unity 
in  the  case  of  a  solvent  which  consists  solely  of  the  kind  of  molecule 
under  consideration,  the  proportionality-factor  in  this  case  is  obvi- 
ously the  vapor-pressure  of  the  pure  solvent.  It  can  be  shown  with 
the  aid  of  the  mass-action  law  that  this  is  true  also  in  the  case  that 
the  solvent  consists  mainly  of  associated  molecules  (such  as  H^) 
and  contains  only  a  small  proportion  of  the  simple  molecules  (such  as 
H2O)  of  which  the  vapor  consists. 

When  the  vapor  contains  two  or  more  kinds  of  molecules,  owing  to 
partial  association  of  the  simple  solvent  molecules  or  to  the  fact  that 
the  solute  has  an  appreciable  vapor-pressure,  it  is  evident  from  these 
considerations  that  p  and  pQ  in  the  Raoult  equation  denote,  not  the 
total  vapor-pressures  of  the  solution  and  solvent  respectively,  but  the 
partial  vapor-pressures  due  to  the  simple  molecules. 

*The  above-given  forms  of  the  Raoult  equation  also  presuppose 
that  the  vapor  conforms  to  the  perfect-gas  law.  Fundamentally  ? 
Raoult's  law  expresses  the  tendency  of  the  solvent  to  escape  from 
the  solution,  and  it  should  therefore  be  independent  of  the  equation 
of  state  of  the  vapor.  It  can  be  shown  by  thermodynamic  considera- 
tions that  the  simple  Raoult  equation  can  be  corrected  for  the  devia- 


VAPOR-PRESSURE  AND  BOILING-POINT  59 

tion  of  the  vapor  from  the  perfect-gas  law  when  this  is  expressed 
(as  in  Art.  15)  by  the  equation  p  v  =  NRT  (i+ap)  by  modifying  it 
by  the  addition  of  another  factor  so  that  it  assumes  the  following 
form,  which  will  be  called  the  corrected  Raoult  equation: 


37.  Relation  of  Boiling-Point  Raising  to  Vapor-Pressure  Lowering 
and  Molal  Composition.  —  A  relation  between  the  increase  in  the 
boiling-point  and  the  decrease  in  the  vapor-pressure  of  a  solvent 
produced  by  a  non-volatile  solute  can  be  derived  from  the  effect  of 
temperature  on  the  vapor-pressure  of  the  solvent  in  the  way  shown 
in  the  following  problems. 

Calculation  of  Boiling-Point  Raising  from  Vapor-Pressure  Lowering.  — 

Prob.  14.  —  a.  What  is  the  vapor-pressure  in  mm.  of  a  solution  at 

100°  containing  5  g.  of  glucose  (C6Hi2O6)  in  100  g.  of  water?    b.  What 

is  its  boiling-point?    Its  vapor-pressure  at  100°,  like  that  of  water, 

increases  3.58%  per  degree. 

Prob.  15.  — The  vapor-pressure  of  ethyl  alcohol  is  721.5  mm.  at  77°, 
751.0  at  78°,  781.5  at  79°,  and  813.0  at  80°.  a.  Plot  on  a  large  scale 
these  vapor-pressures  as  ordinates  and  the  temperatures  as  abscissas. 
Calculate  the  vapor-pressures  at  78,  79,  and  80°  of  a  solution  consist- 
ing of  2  mols  of  a  non-volatile  solute  and  98  mols  of  alcohol,  and  of  one 
consisting  of  4  mols  of  solute  and  96  mols  of  alcohol;  and  plot  these  values 
on  the  diagram,  b.  With  the  aid  of  the  diagram  find  the  boiling-points 
at  i  atm.  of  pure  alcohol  and  of  the  two  solutions,  c.  Show  from  the 
geometrical  relations  of  the  diagram  that  the  raising  of  the  boiling- 
point  is  proportional  to  the  lowering  of  the  vapor-pressure  at  the  boil- 
ing-point of  the  solvent,  in  the  case  of  dilute  solutions  (for  which  the 
graphs  may  be  considered  to  be  parallel  straight  lines),  d.  Show  from 
the  geometrical  relations  that  the  ratio  of  the  lowering  of  the  vapor- 
pressure  to  the  raising  of  the  boiling-point,  caused  by  adding  the  solute, 
is  equal  to  the  ratio  of  the  increase  in  the  vapor-pressure  of  the  solvent 
caused  by  increasing  the  temperature  to  the  increase  in  temperature, 
in  the  case  of  dilute  solutions  (for  which  the  graphs  may  be  considered 
straight  lines). 

Representing  by  T—  TQ  the  raising  of  the  boiling-point,  and  by 
pQ-p  the  lowering  of  the  vapor-pressure,  produced  by  increasing  the 
mol-fraction  of  the  solute  from  o  to  x,  and  representing  by  A/>0/A7Ythe 
rate  of  change  of  the  vapor-pressure  of  the  solvent  with  the  tempera- 
ture, the  relations  derived  in  the  preceding  problem  for  dilute  solu- 
tions may  be  expressed  by  the  equation: 


A7Y 


60  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

A  T*  ^7^7^ 

The  quantity  .  A  ,°    evidently  corresponds  to  the  quantity  -j—n 
*  bpo/po  dp/p 

in  the  approximate  Clapeyron  equation,  and  it  is  therefore  equal  to 

7?T"  ^  ~ 

—=r-  ,  where  A#0  denotes  the  molal  heat  of  vaporization  of  the  solvent 

A#0 

at  its  boiling-point  TV    The  relation  between  boiling-point  raising 

and  fractional  vapor-pressure  lowering  is  therefore  also  given  approxi- 

mately by  the  expression: 


By  combining  the  expressions  just  derived  with  Raoult's  equation 
for  vapor-pressure  lowering  there  may  evidently  be  obtained  the 
following  relations  between  the  raising  of  the  boiling-point  (T—T0) 
and  the  mol-fraction  x  of  the  solute: 

/  dT0  \        (RTf\  () 

T~T<°  \dpjpj*-  \-JnJ  *• 

*The  preceding  equations  become  exact  only  when  infinitesimal 
changes  hi  temperature,  vapor-pressure,  and  mol-fraction  are  con- 
sidered; for  only  in  that  case  are  the  graphs  in  the  diagram  of  Prob.  15 
from  which  these  equations  were  derived  strictly  parallel  straight 
lines.  It  will  be  seen  from  the  diagram  that  the  following  differential 
equation,  corresponding  to  equation  (i)  above,  holds  true  for  any 
solution  with  a  non-  volatile  solute  having  any  mol-fraction  x: 

dT\          (dT\    (dp\  . 

/,~  "V^/.Vte/r 
*From  this  equation  there  can  be  derived,  as  shown  in  Prob.  16,  by 
substituting  for  the  last  two  partial  derivatives  the  expressions  for 
them  given  by  the  Clapeyron  equation  (Art.  33)  and  the  Raoult  equa- 
tion (Art.  36),  the  following  equation: 

RT*     dx  M 

dT=~^s  (^r  (5) 

In  this  equation  dT  represents  the  raising  in  the  boiling-point  T 
caused  by  increasing  the  mol-fraction  of  the  solute  from  x  to  x+dx, 
and  A#  represents  the  heat  absorbed  in  vaporizing  one  mol  of  solvent 
at  the  temperature  T  out  of  an  infinite  quantity  of  the  solution  (since 
it  can  be  shown  to  have  this  significance  when  the  Clapeyron  equation 
is  applied  to  vaporization  out  of  a  solution). 


VAPOR-PRESSURE  AND  BOILING-POINT  61 

*Prob.  16.  —  Boiling-Point  of  Solutions  Conforming  to  RaouWs  Law 
at  Any  Concentration.  —  a.  State  in  words  what  each  of  the  partial 
derivatives  in  equation  (4)  signifies,  b.  Find  from  the  exact  Clapeyron 
equation  (neglecting  the  volume  of  the  liquid)  and  from  the  corrected 
Raoult  equation  (given  at  the  end  of  Art.  36)  expressions  for  the  last 
two  of  these  partial  derivatives,  respectively,  c.  By  substituting 
these  expressions  in  the  partial  derivative  equation,  and  simplifying 
it  with  the  aid  of  the  equation  of  state  pv  =  RT  (i-f  ap),  derive  the 
expression  for  the  boiling-point  raising  formulated  as  equation  (5). 

*Equation  (5)  holds  true  even  when  the  saturated  vapor  of  the 
solvent  does  not  conform  to  the  perfect-gas  law,  as  shown  by  its 
derivation.  It  can  be  integrated  (as  in  Prob.  17),  usually  with 
negligible  error,  under  the  assumption  that  A#  does  not  vary  with 
the  temperature  or  with  the  mol-fraction  of  the  solute,  and  that 
therefore  it  has  the  same  value  as  the  molal  heat  of  vaporization  A&0 
of  the  pure  solvent  at  its  boiling-point. 

*Prob.  17.  —  Integration  of  the  Boiling-Point  Equation.  —  a.  Inte- 
grate equation  (5)  given  in  the  text,  assuming  that  the  heat  of  vaporiza- 
tion does  not  vary  with  the  temperature  or  the  mol-fraction  of  the 
solute,  so  as  to  obtain  a  relation  between  the  boiling-point  of  the  solu- 
tion, the  boiling-point  of  the  pure  solvent,  and  the  mol-fraction  of  the 
solute,  b.  Calculate  the  boiling-point  of  a  solution  consisting  of  10 
mols  of  a  non-volatile  solute  and  90  mols  of  benzene.  The  heat  of 
vaporization  of  one  gram  of  benzene  at  its  boiling-point  80.3°  is  93.0  cal. 

*For  small  values  of  x  and  for  the  correspondingly  small  changes 
hi  the  boiling-point  T  equation  (5)  may  be  integrated  under  the 
assumption  that  i  —  x  is  equal  to  unity  and  that  T  is  constant  and 
equal  to  To,  and  there  then  results  equation  (3),  previously  derived 
for  dilute  solutions,  namely: 

T      T       RT°\ 

1  —  1  o  =  —  —  x. 


jrrt  7PT1  2 

The  quantity  —    -  or  —  =?•  occurring  in  equation  (3)  is  evidently 


a  constant  characteristic  of  the  solvent.  Representing  this  quantity, 
which  may  be  called  the  boiling-point  constant,  by  a  single  letter  B0, 
and  noting  that  the  mol-fraction  x  approaches  the  mol-ratio  N/  N0  as 
its  value  approaches  zero,  the  law  of  boiling-point  raising  for  perfect 
solutions  may  be  expressed  by  the  equation: 

T-T^B^.  (6) 

No 


62  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

It  is  evident  from  this  equation  that  the  boiling-point  constant 
BQ  is  the  ratio  of  the  boiling-point  raising  to  the  number  of  mols  of 
solute  which  are  associated  with  one  mol  of  solvent;  or,  briefly,  it  is 
the  boiling-point  raising  per  mol  of  solute  in  one  mol  of  solvent.  It 
can  be  experimentally  determined  by  measuring  the  rise  in  boiling- 
point  produced  by  dissolving  a  definite  weight  of  a  solute  of  known 
molecular  weight  in  a  definite  quantity  of  solvent.  It  can  also,  of 
course,  be  calculated  from  the  change  of  vapor-pressure  of  the  solvent 
with  the  temperature  or  from  the  molal  heat  of  vaporization  of  the 
solvent  by  the  above-given  expressions.  There  are,  accordingly, 
three  kinds  of  data  with  the  aid  of  which  the  boiling-point  constant 
can  be  evaluated,  as  illustrated  in  Prob.  18. 

In  chemical  literature  is  commonly  recorded,  not  this  boiling-point 
constant,  but  another  constant,  called  the  mold  boiling-point  raising, 
which  is  the  boiling-point  raising  per  mol  of  solute  in  1000  grams  of 
solvent.  The  value  of  this  constant  can  obviously  be  readily  calcu- 
lated from  that  of  the  boiling-point  constant  (as  in  Prob.  19). 

Prob.  18.  —  Methods  of  Determining  the  Boiling-Point  Constant.  — 
Calculate  the  boiling-point  constant  for  ethyl  alcohol  from  the  following 
data.  a.  The  heat  of  vaporization  of  one  gram  is  206  cal.  at  the 
boiling-point  78.3°.  b.  Its  vapor-pressure  has  the  values  given  in 
Prob.  15.  c.  The  boiling-point  of  a  solution  of  i  g.  of  naphthalene 
(CioH8)  in  50  g.  of  alcohol  is  0.185°  higher  than  that  of  pure  alcohol. 

Prob.  19.  —  Relation  between  the  Boiling-Point  Constant  and  the  Molal 
Boiling-Point  Raising.  —  a.  Calculate  the  molal  boiling-point  raising 
for  water  from  its  boiling-point  constant,  b.  Formulate  the  algebraic 
relation  between  the  two  constants. 

The  following  table  contains  the  values  of  the  constants  for  a  few 
important  solvents. 

BOILING-POINT  CONSTANTS 

Water       Ethyl  ether    Ethyl  alcohol    Benzene 

mo       CiHioO       amon       c&Ht 


Boiling-point  constant       .      .     28.6  28.5          25.8        34.0 

Molal  boiling-point  raising      .       0.515         2.11          1.19       2.65 

38.  Determination  of  Molecular  Weights.  —  From  the  value  of 
either  of  the  boiling-point  constants  for  a  given  solvent  there  may  be 
calculated  by  direct  proportion  the  number  of  mols  4of  solute  corre- 
sponding to  any  observed  raising  of  the  boiling-point.  The  law  of 
boiling-point  raising,  like  Raoult's  law  from  which  it  has  been  derived, 
therefore  makes  it  possible  to  determine  the  molecular  weights  of 
substances  in  solution. 


VAPOR-PRESSURE  AND  BOILING-POINT  63 

Determination  of  Molecular  Weight  and  Molecular  Composition.  — 
Prob.  20.  —  a.  When  10.6  g.  of  a  substance  are  dissolved  in  740  g.  of 
ether  (C4Hi0O),  its  boiling-point  is  raised  0.284°.     What  is  the  molec- 
ular weight   of   the  substance?     b.  The  substance  is  a  hydrocarbon 
containing  90.50%  of  carbon.    What  is  its  molecular  formula? 

Prob.  21. — •  A  solution  of  3.04  g.  of  benzoic  acid  in  100  g.  of  ethyl 
alcohol  boils  0.288°  higher  than  pure  alcohol.  A  solution  of  6.34  g.  of 
benzoic  acid  in  100  g.  of  benzene  boils  0.696°  higher  than  pure  benzene. 
Calculate  the  molecular  weight  of  benzoic  acid  in  each  of  these  solvents, 
and  state  what  the  results  show  in  regard  to  its  molecular  formula  in 
each  solvent.  Its  composition  by  weight  is  expressed  by  C6H5CO2H. 

The  molecular  weights  of  substances  are  ordinarily  found  to  be  the 
same  in  the  dissolved  state  as  in  the  gaseous  state;  but  hydroxyl  com- 
pounds (such  as  the  alcohols  and  organic  acids)  form  in  non-oxygen- 
ated solvents  (such  as  benzene  or  chloroform)  double  or  even  more 
highly  associated  molecules.  This  indicates  that  the  molecules  of  hy- 
droxyl compounds  are  associated  also  in  the  state  of  pure  liquids. 
Oxygenated  solvents  (such  as  water,  alcohols,  acetic  acid,  etker,  and 
acetone)  have  the  power  of  breaking  down  these  associated  molecules 
into  the  simple  ones;  and  in  such  solvents  nearly  all  organic  substances 
have  the  same  molecular  weight  as  in  the  gaseous  state. 

39.   Partial  Vapor-Pressure  of  Volatile  Solutes.     Henry's  Law.  — 

In  addition  to  Raoult's  law,  which  relates  to  the  vapor-pressure  of 
the  solvent  in  a  perfect  solution,  there  is  another  fundamental  law 
which  relates  to  the  vapor-pressure  of  the  solute  in  such  a  solution. 
This  law,  known  as  Henry's  Law,  may  be  stated  as  follows.  The 
partial  vapor-pressure  of  any  chemical  substance  present  in  small 
proportion  in  a  solution  is  proportional  to  its  mol-fraction.  That  is, 

p  =kx, 

where  k  is  a  proportionality-constant  dependent  on  the  nature  of  the 
solute  and  of  the  solvent  and  on  the  temperature. 

Prob.  22. — Application  of  Henry's  Law  and  Raoult's  Law. — The 
total  vapor-pressure  of  a  solution  containing  3.00%  by  weight  of  ethyl 
alcohol  (C2H5OH)  in  water  is  760  mm.  at  97.11°,  and  the  vapor-pressure 
of  pure  water  at  this  temperature  is  685  mm.  Calculate  with  the 
help  of  Raoult's  law  and  of  Henry's  law  the  vapor-pressures  at  97.11° 
of  ethyl  alcohol  and  water  in  a  solution  containing  2.00  mol-percent  tf 
ethyl  alcohol.  Ans.  139  mm.  for  C2H8OH. 

Henry's  law  may  also  be  expressed  in  the  following  form,  which 
makes  it  more  obvious  that  it  is  a  law  of  equilibrium  which  determines 
the  distribution  of  the  solute  between  the  solution  and  a  gaseous 


64  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

phase,  and  which  incidentally  expresses  the  composition  of  the  solution 
in  terms  either  of  molality  or  of  molal  concentration  (denned  as  in  Art. 
35),  instead  of  as  mol-fraction.  Any  chemical  substance  present  as  a 
perfect  gas  in  a  vapor  phase  and  as  a  solute  in  a  perfect  solution  in 
equilibrium  with  it  has  at  any  definite  temperature  a  molality  or 
molal  concentration  c  in  the  solution  which  is  proportional  to  its 
(partial)  pressure  p  in  the  gaseous  phase.  That  is, 

c/p=K, 

where  K  is  an  equilibrium-constant  which  is  determined  by  the  nature 
of  the  chemical  substance  and  of  the  solvent  and  by  the  temperature. 
The  equilibrium  concentration  c  is  the  solubility  of  the  substance  when 
its  partial  pressure  in  the  gas  phase  is  p,  and  the  equilibrium- 
constant  K  may  be  called  the  solubility-constant  of  the  gaseous  sub- 
stance in  the  solvent.  Henry's  law  is  therefore  a  law  of  the 
solubility  of  gases. 

This  second  form  of  Henry's  law  is  for  perfect  solutions  equivalent 
to  the  first  form.  For,  on  the  one  hand,  the  mol-fraction  N/  NQ+  N), 
as  it  approaches  zero,  becomes  equal  to  the  mol-ratio  N/  No,  and  this 
is  evidently  proportional  to  the  molality  (ioooN/  m0)  or  to  the  molal 
concentration  ( N/vQ) ;  and,  on  the  other  hand,  the  vapor-pressure  of  a 
substance  in  a  solution  is  equal  to  its  partial  pressure  iri  a  gaseous 
phase  that  is  in  equilibrium  with  the  solution.  / 

Henry's  law  in  either  of  its  two  forms  is  conformed  to  more  closely 
as  the  pressure  of  the  gas  and  the  concentration  of  the  solute  approach 
zero.  Like  the  other  laws  of  perfect  solutions,  it  usually  holds  true 
within  2  to  3  percent,  even  when  the  pressure  is  one  atmosphere  and 
the  concentration  i  molal. 

It  is  to  be  noted  that  Henry's  law  expresses  conditions  of  equi- 
librium, and  that  these  conditions  are  often  attained  between  a  gaseous 
and  liquid  phase  only  by  long-continued  intimate  contact. 

From  a  molecular  standpoint,  Henry's  law,  like  Raoult's  law,  relates 
to  the  distribution  of  some  definite  kind  of  molecule  between  the  gas 
phase  and  the  liquid  phase.  Hence  in  applications  of  it  the  same 
chemical  substance  in  the  two  phases  must  be  considered.  Thus,  when 
the  chemical  substance  SO*  dissolves  in  water  it  is  largely  converted 
into  H2SO3  and  its  ions  H+  and  HSO3~;  and  Henry's  law  therefore 
requires,  not  that  the  total  concentration  of  solute  in  the  solution,  but 
that  the  concentration  of  the  SO2  itself,  be  proportional  to  the  partial  J 
pressure  of  the  S02  in  the  vapor.  When,  however,  the  only  change  in  / 


VAPOR-PRESSURE  AND  BOILING-POINT  65 

the  substance  is  that  it  partially  combines  with  the  solvent  forming  a 
solvate  (a  hydrate  in  the  case  of  water),  then  the  total  concentration 
may  be  employed;  for  the  fraction  solvated  is  in  dilute  solution  in- 
dependent of  the  concentration  of  the  substance,  as  may  be  shown 
by  the  mass-action  law.  Thus,  though  the  substance  CO2  on  dissolv- 
ing in  water  is  partly  converted  into  the  hydrate  H2CO3  (which  is  sub- 
stantially unionized,  except  at  very  small  concentrations),  yet  the 
solubility  of  carbon  dioxide  gas,  as  found  by  determining  the  total 
quantity  dissolved,  changes  with  the  pressure  in  accordance  with 
Henry's  law. 

These  considerations  also  show  that  in  applying  Henry's  law  the 
partial  pressure  of  the  chemical  substance  in  the  gas  phase,  not  the  total 
pressure  of  the  gas,  must  be  considered.  Thus  the  quantity  of  carbon 
dioxide  dissolved  by  water  in  contact  with  air  is  not  determined  by  the 
pressure  of  the  air,  but  by  the  partial  pressure  of  CO-2  in  the  air. 

Prob.  23.  —  Determination  of  the  Vapor-Pressure  of  Solutes.  —  A 
mixture  of  air  and  ammonia  containing  i  mol-percent  of  NH3  is  passed 
at  25°  and  i  atm.  through  water.  The  saturated  solution  is  found  by 
titration  to  be  0.553  formal  in  NH4OH.  Calculate  the  partial  vapor- 
pressure  of  NH3  in  a  i  formal  solution  at  25°.  The  vapor-pressure  of 
water  at  25°  is  23.8  mm. 

Solubility  of  Gases  in  Water.  — 

Prob.  24.  —  In  a  gas  buret  over  mercury  60  ccm.  of  dry  carbon 
dioxide  at  25°  and  i  atm.  are  placed,  40  ccm.  of  water  are  introduced, 
and  the  gas  and  water  are  shaken  together  at  25°  till  equilibrium  is 
reached,  keeping  the  pressure  on  the  gas  i  atm.  The  volume  of  the 
(moist)  gas  is  then  found  to  be  28.9  ccm.  Calculate  the  molal  solubility 
of  carbon  dioxide  in  water  at  25°  when  its  partial  pressure  is  i  atm., 
neglecting  effects  that  influence  the  result  less  than  0.5%.  Ans.  0.0338. 

Prob.  25.  At  20°  100  g.  of  water  dissolve  3.4  ccm.  of  oxygen,  1.7  ccm. 
of  nitrogen,  and  3.8  ccm.  of  argon  when  the  pressure  of  each  gas  is  i  atm. 
a.  Calculate  the  corresponding  molal  solubility  of  each  gas.  b.  Calcu- 
late the  mol-fraction  of  each  constituent  in  the  gas-mixture  obtained 
by  shaking  water  at  20°  with  air  (free  from  CO2) ,  expelling  the  dissolved 
gas  by  boiling  and  drying  it.  Tabulate  the  molal  composition  of  this 
gas  with  that  of  air  (given  in  Art.  12).  Ans.  b,  0.34  for  O2. 

Prob.  26.  —  Determination  of  the  State  of  Substances  in  Solution.  — 
The  partial  vapor-pressure  of  NH3  in  an  aqueous  solution  0.3  formal  in 
NH3  and  o.i  formal  in  AgNO3  is  at  25°  equal  to  that  in  a  o.i  formal 
solution  of  NH3  in  water.  State  and  explain  the  conclusion  that  can 
be  drawn  as  to  the  formula  of  the  complex  salt  formed,  considering  all 
the  silver  nitrate  to  be  combined  with  ammonia,  and  assuming  that  the 
solubility-constant  of  the  NH3  is  not  affected  by  the  silver  salt. 


66  THE   MOLAL  PROPERTIES  OF  SOLUTIONS 

Aside  from  any  chemical  action,  the  addition  to  the  water  of  a  salt 
commonly  decreases  the  solubility  of  a  volatile  substance  at  a  given 
pressure  in  the  gas  phase;  or  conversely,  it  increases  its  vapor-pressure 
at  a  given  concentration  in  the  solution.  This  phenomenon,  which  is 
known  as  the  salting-out  effect,  is  subject  to  the  following  principles: 

(1)  The  decrease  of  the  solubility-constant  is  approximately  pro- 
portional to  the  concentration  of  the  added  salt,  up  to  concentrations 
not  much  exceeding  i  normal. 

(2)  The  fractional  decrease  per  equivalent  of  salt  per  1000  grams  of 
water  is  roughly  the  same  for  a  definite  salt,  whatever  be  the  nature  of 
the  solute. 

(3)  This  fractional  decrease  varies  greatly  with  the  nature  of  the 
salt;  thus  the  decrease  caused  by  o.i  equivalent  of  salt  per  1000 
grams  of  water  varies  from  about  zero  in  the  case  of  barium  nitrate  to 
about  4  percent  in  the  case  of  potassium  and  sodium  sulfates. 

Prob.  27.  —  The  Salting- out  Effect.  —  The  solubility  (that  is,  the 
concentration  of  the  saturated  solution)  of  carbon  dioxide  at  25°  and 
i  atm.  is  0.0338  molal  in  pure  water  and  0.0331  in  a  i  normal  NaCl 
solution.  The  vapor-pressure  of  ammonia  from  a  0.5  molal  solution  of 
it  in  water  at  25°  is  6.65  mm.  Predict  from  the  principles  of  the  salting- 
out  effect  the  ammonia  vapor-pressure  for  a  solution  0.5  molal  in  NH3 
and  0.5  normal  in  NaCl. 


VAPOR-PRESSURE  AND  BOILING-POINT  67 

IV.    VAPOR-PRESSURE   AND    BOILING-POINT    OF    CONCENTRATED 
SOLUTIONS   WITH   TWO   VOLATILE   COMPONENTS 

40.  The  Vapor-Pressure  of  Concentrated  Solutions. —  Concentrated 
solutions  may  be  divided  for  purposes  of  consideration  into  two  groups 
as  follows.  One  group  consists  of  those  solutions  whose  formation 
out  of  their  pure  components  (when  these  are  liquid)  is  not  attended 
by  any  considerable  change  of  temperature  or  volume,  and  whose 
'properties  in  general  are  approximately  the  sum  or  average  of  those 
of  the  pure  components.  The  characteristic  of  such  solutions  is  that 
neither  component  exerts  a  specific  influence  on  the  properties  of  the 
other  component.  Such  solutions  conform  approximately  —  more 
closely  as  the  condition  characterizing  them  is  more  nearly  fulfilled  — 
to  the  laws  of  perfect  solutions.  Concentrated  solutions  which  are 
considered  to  conform  to  these  laws  may  be  called  concentrated  perfect 
solutions.  The  other  group  of  concentrated  solutions  consists  of  those 
whose  components  exert  a  marked  influence  upon  one  another.  For 
these  solutions  no  general  laws  are  known. 

The  method  commonly  employed  for  experimentally  determining 
at  any  definite  temperature  the  partial  vapor-pressures  of  solutions 
consisting  of  two  or  more  volatile  components  is  to  distil  off  a  small 
fraction  from  a  large  volume  of  the  solution,  adjusting  the  pressure 
on  the  liquid  so  that  it  boils  at  this  temperature.  The  composition 
of  the  distillate  is  then  determined  by  chemical  analysis  or  by  the 
measurement  of  some  physical  property,  such  as  density;  and  from 
this  composition  and  the  pressure  under  which  the  distillation  took 
place  the  partial  vapor-pressures  are  calculated  by  Dalton's  law. 

Prob.  28. —  Experimental  Determination  of  Partial  Vapor-Pressures. — 
A  solution  of  two  substances  AandB  containing  N\  mols  of  A  and  NB 
mols  of  B  boils  at  the  temperature  T  when  a  pressure  p  is  exerted 
upon  it.  The  first  portion  of  distillate  consists  of  NAf  mols  of  A  and 
N*  mols  of  B.  Derive  an  algebraic  expression  for  the  partial  vapor- 
pressures  pA  and  pv  of  the  two  substances  in  the  solution,  explaining 
the  principles  involved. 

41.  Vapor-Pressure  and  Boiling-Point  of  Concentrated  Perfect 
Solutions  in  Relation  to  their  Molal  Composition.  —  In  concentrated 
perfect  solutions  the  vapor-pressure  of  each  component  conforms 
approximately  to  Raoult's  law.  In  other  words,  the  partial  vapor- 
pressure  of  each  component  is  approximately  equal  to  the  product 
of  its  mol-fraction  in  the  solution  by  its  vapor-pressure  in  the  pure 
state,  whatever  be  the  proportion  in  which  the  components  are  present; 
that  is,  pA  =  pQA  XA,  P* 


68  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

Prob.  2Q.  —  Vapor-Pressure-Composition  Diagram  for  Perfect  Solu- 
tions. —  At  50°  the  partial  vapor-pressures  of  benzene  and  of  ethylene 
chloride  in  solutions  of  these  two  substances  have  been  found  experi- 
mentally to  have  the  following  values: 

Mol-fraction  Vapor-pressure  Vapor-pressure 

ofCtHe  o/CtHs  of  CzHtCk 

i.ooo  268  mm.  o  mm" 

0.707  190  68 

0.478  128  123 

0.246  66  178 

o.ooo  o  236 

a.  Plot  on  a  large  scale  these  partial  vapor-pressures,  and  the  corre- 
sponding total  vapor-pressures,  as  ordinates  against  the  mol-fractions 
as  abscissas.  Show  that  the  three  graphs  are  in  almost  complete  accord 
with  Raoult's  law.  b.  Calculate  the  mol-fraction  of  benzene  in  the 
vapor  which  at  50°  is  in  equilibrium  with  each  of  the  three  solutions 
for  which  the  data  are  given  in  the  above  table,  c.  On  a  new  larger- 
scale  vapor-pressure-composition  diagram  (with  ordinates  covering  only 
the  interval  of  230-270  mm.)  draw  a  line  showing  the  variation  of  the 
total  vapor-pressure  with  the  mol-fraction  of  the  liquid  mixture.  Plot 
also  on  this  diagram  the  compositions  of  the  vapor  calculated  in  b 
against  the  total  vapor-pressures,  considering  that  the  abscissas  now 
represent  the  mol-fraction  of  benzene  in  the  vapor; 

Prob.  30.  —  Distillation  at  Constant  Temperature  of  Perfect  Solutions. — 
At  50°  a  small  fraction  is  distilled  off  from  a  large  volume  of  a  solu- 
tion containing  equimolal  quantities  of  benzene  and  ethylene  chloride, 
and  this  distillate  is  redistilled  at  50°.  Derive  from  the  diagram  of 
Prob.  29  the  mol-fraction  of  benzene  in  the  first  part  of  the  second 
distillate.  Ans.  0.56. 


*The  boiling-point  of  a  solution  that  contains  two  volatile  components 
whose  partial  vapor-pressures  both  conform  to  Raoult's  law  can  best 
be  derived  graphically  from  the  vapor-pressures  of  the  pure  substances, 
as  illustrated  by  the  following  problem. 

*Prob.  31. —  Boiling-Point-Composition  Diagram  for  Perfect  Solutions. 
The  vapor  pressure  at  ....     80°,     83°,     86°,     89°,     92° 
of  a  pure  liquid  A  is  .      .     .      .     560,     610,     665,     725,     790  mm. 
and  of  a  pure  liquid  B  is      .     .     400,    435,    475,     520,     570  mm. 

a.  With  the  aid  of  these  data  and  Raoult's  law,  draw  on  a  large-scale 
vapor-pressure-composition  diagram  for  each  temperature  two  lines  — 
one  representing  the  total  vapor-pressure  of  any  solution  of  A  and  B ; 
and  the  other  representing  the  partial  vapor-pressure  of  A  above  any 
solution,  b.  Determine  from  the  plot  the  composition  of  the  liquid 
which  at  570  mm.  boils  at  each  of  these  temperatures;  also  the  composi- 
tion of  the  vapor  which  is  in  equilibrium  with  each  of  these  solutions  at 


J 


VAPOR-PRESSURE  AND  BOILING-POINT  69 

its  boiling-point.  Determine  also  the  boiling-point  of  the  pure  liquid  A 
at  570  mm.,  and  tabulate  all  of  these  results,  c.  On  another  large- 
scale  diagram  plot  against  these  liquid-compositions  the  boiling-points 
as  ordinates.  Plot  also  the  vapor-compositions  against  the  correspond- 
ing boiling-points. 

Distillation  at  Constant  Pressure  of  Perfect  Solutions.  — 
*Prob.  32.  —  A  solution  of  100  mols  of  each  of  the  liquids  of  Prob.  31  is 
distilled  at  570  mm.  until  its  boiling-point  rises  0.5°.  a.  Find  from  the 
diagram  of  Prob.  31^  the  molal  compositions  of  the  first  and  last  por- 
tions of  the  distillate,  b.  Regarding  the  composition  of  the  whole 
distillate  as  the  mean  of  that  of  its  first  and  last  portions  (which  is 
approximately  true  when  only  a  small  fraction  of  the  liquid  distils  over), 
calculate  the  number  of  mols  of  A  and  of  B  in  the  distillate  and  in  the 
residue,  c.  The  distillation  of  the  residue  is  continued  till  its  boiling- 
point  rises  0.5°  more.  Calculate  as  in  b  the  number  of  mols  of  A  and 
B  in  this  second  distillate  and  in  the  residue,  d.  Tabulate  the  number 
of  mols  of  A  and  B  in  the  original  liquid,  the  first  distillate,  the  second 
distillate,  and  the  final  residue;  the  mol-fraction  of  A  in  each  of  these 
liquids;  and  the  boiling-point  of  each  of  them.  Ans.  b,  52  mols  of  A, 
and  63  mols  of  B  in  the  residue. 

*Prob.  33.  a.  The  first  distillate  obtained  in  Prob.  320  is  redistilled 
until  the  residue  attains  the  composition  of  the  second  distillate  ob- 
tained in  Prob.  32^.  Find  the  mol-fraction  of  A  in  the  new  distillate  and 
its  boiling-point,  b..  The  residue  is  now  mixed  with  the  second  distil- 
late obtained  in  Prob.  32^,  and  the  distillation  is  continued  till  the 
residue  has  the  composition  of  the  residue  obtained  in  Prob.  32^.  Find 
the  mol-fraction  of  A  in  the  distillate  thus  obtained  and  its  boiling- 
point,  c.  Tabulate  the  composition  and  the  boiling-point  of  the 
original  equimolal  solution  and  of  the  three  fractions  into  which  it  has 
now  been  resolved.  Ans.  b,  mol-fraction,  0.55;  boiling-point,  85.5°. 

The  foregoing  considerations  show  that  any  perfect  solution  sub- 
mitted to  distillation  resolves  itself  into  a  distillate  containing  a 
larger  proportion,  and  into  a  residue  containing  a  smaller  proportion, 
of  the  more  volatile  component.  It  is  evident  that,  in  consequence 
of  this  behavior,  the  two  components  can  be  completely  separated 
from  each  other  by  repeated  fractional  distillation,  carried  out  as 
illustrated  by  Probs.  32  and  33. 

42.  Vapor-Pressure  of  Concentrated  Solutions  in  General  in  Rela- 
tion to  their  Composition.  —  There  are  comparatively  few  actual 
solutions  which  fulfil  strictly  the  criterion  stated  in  Art.  40  of  being 
formed  out  of  their  components  without  any  change  of  temperature 
or  volume;  and  correspondingly,  comparatively  few  concentrated 


70  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

solutions  conform  completely  to  Raoult's  law  of  perfect  solutions. 
The  law  is  therefore  to  be  regarded  as  a  limiting  law,  from  which  actual 
solutions  deviate  to  an  extent  which  is  as  a  rule  roughly  indicated 
by  the  magnitude  of  the  changes  of  temperature  and  volume  attending 
the  mixing  of  the  components. 

The  data  given  below  illustrate  the  magnitude  of  the  deviations 
from  the  law  for  a  variety  of  solutions.  The  first  two  columns  of 
figures  show  the  change  of  temperature  and  the  percentage  change 
of  volume  which  result  when  equimolal  quantities  of  the  two  sub- 
stances at  the  same  temperature  are  mixed.  The  last  three  columns 
show  the  percentage  difference  between  the  observed  values  of  the 
partial  (or  total)  vapor-pressures  of  the  equimolal  solution  and  those 
calculated  by  Raoult's  law;  namely,  the  values  of  the  vapor-pressure 

ratio  I00(£obs. — £calc.)//>calc. 

DEVIATIONS  FROM  RAOULT'S  LAW 

Mixture  Components  Change  of     Change  of          Percentage  Deviation  of  Pressure 

No.  A  B  Temperature      Volume  A  B  A+B 

1  C6H6C1  C6H6Br  0.0°       0.0%  -  o.i  -  o.i 

2  C6H6      C2H4C12  —  0.3  +0.34  +0.1  —  0.2 

3  CHC13   (CH3)2CO  +12.4  -0.23  -23.  -18. 

4  CS2        CH2(OCH8)2  -  6.5  +1.22  +10.  +  7. 

5  CSa        (CH3)2CO  -  9.8  +i.2i  +40.  +58. 

6  C6H6      C2H6OH  -  4.2        o.o  +50 

The  character  of  the  deviations  from  Raoult's  law  can  be  best 
appreciated  by  considering  their  causes  from  molecular  and  kinetic 
view-points.  It  has  already  been  seen  (in  Art.  34)  that  the  vapor- 
pressure  of  a  solvent  is  determined  primarily  by  the  proportion  of  its 
molecules  which  at  any  temperature  acquire  an  outward  velocity 
sufficient  to  overcome  the  inward  attraction  of  the  other  molecules 
and  thus  escape  through  the  surface  layer  into  the  gas  phase.  Con- 
sider now  the  different  conditions  that  may  result  when  another  sub- 
stance B  is  added  to  the  solvent  A. 

(i)  In  case  the  B  molecules  exert  upon  the  A  molecules  the  same 
attraction  as  the  A  molecules  exert  on  one  another,  the  attraction 
upon  the  A  molecules  in  the  mixture,  and  hence  their  escaping  velocity, 
will  be  the  same  as  in  the  pure  solvent,  the  proportion  of  them  that 
acquire  this  escaping  velocity  will  be  the  same,  and  the  absolute 
number  of  them  which  escape  will  be  diminished  only  in  the  proportion 
in  which  the  proportion  of  A  molecules  in  the  liquid  is  reduced.  In 
this  case,  therefore,  the  vapor-pressure  of  A  will  be  directly  proper- 


VAPOR-PRESSURE  AND  BOILING-POINT  71 

tional  to  its  mol-fraction,  as  Raoult's  law  requires;  and  little  if  any 
heat-effect  or  volume  change  will  take  place  on  mixing  the  components. 

(2)  In  case  the  B  molecules  exert  upon  the  A  molecules  a  smaller 
attraction  than  they  exert  upon  one  another,  the  escaping  velocity 
will  be  smaller  and  the  proportion  of  the  A  molecules  that  acquire 
this  velocity  will  be  greater  in  the  solution  than  in  the  pure  solvent; 
and  correspondingly  the  vapor-pressure  of  A  will  be  greater  than 
Raoult's  law  requires.     In  this  case  the  mixing  of  the  two  components 
will  be  attended  by  an  increase  in  volume  and  a  fall  in  temperature 
(due  to  the  expansion  and  attendant  separation  of  the  molecules  in 
opposition  to  their  attraction). 

(3)  In  case  the  B  molecules  exert  upon  the  A  molecules  a  greater 
attraction  than  the  latter  exert  on  one  another,  the  opposite  effects 
to  those  just  considered  will  evidently  result. 

(4)  In  case  the  B  molecules  form  with  the  A  molecules  a  chemical 
compound  (such  as  AB,  A2B,  or  AB2),  the  fraction  of  A  molecules  in 
the  liquid  is  decreased,  not  only  as  always  by  the  presence  of  the  new 
molecules,  but  in  this  case  also  by  the  conversion  of  some  of  the  A 
molecules  into  those  of  the  compound.     This  will  evidently  result  in  a 
smaller  vapor-pressure  than  Raoult's  law  requires.    In  this  case  an 
evolution  of  heat  and  diminution  of  volume  may  be  expected,  since 
the  synthetic  formation  of  chemical  compounds  is  usually  attended 
by  these  effects. 

(5)  In  case  the  simple  molecules  A  of  the  solvent  are  partially 
associated,  to  form  double  or  more  complex  molecules,  such  as  Aa  or 
A3,  some  of  the  latter  will  be  dissociated,  in  accordance   with  the 
mass-action  law,  when  the  solvent  is  diluted  by  the  addition  of  the 
substance  B,  and  the  partial  pressure  of  the  simple  A  molecules  in  the 
vapor  will  be  correspondingly  increased.   In  this  case,  a  fall  in  tempera- 
ture may  be  expected,  since  dissociation  is  commonly  attended  by 
absorption  of  heat. 

The  foregoing  considerations  are  not  of  a  quantitative  character; 
and  on  the  chemical  effects  described  in  (4)  and  (5)  may  be  super- 
posed the  physical  effects  mentioned  in  (2)  and  (3).  These  molecular 
considerations,  however,  often  assist  in  interpreting  the  vapor-pressure 
of  mixtures. 

Prob.  34.  —  Molecular  Explanations  of  the  Deviations  from  Raoult's 
Law.  —  Suggest  the  most  probable  molecular  explanation  of  the  be- 
havioj  of  each  of  the  mixtures  for  which  data  are  given  in  the  above 
table,  taking  into  account  all  the  foregoing  considerations. 


72  THE   MOLAL  PROPERTIES  OF  SOLUTIONS 

Prob.  55.  —  Compound  Formation  Derived  from  the  Vapor-Pressure 
Relations. — At  35.2°  the  vapor-pressure  of  acetone,  (CH3)2CO,  is 
345  mm.  and  that  of  chloroform,  CHC13,  is  293  mm.  In  a  50  mol- 
percent  mixture  of  these  substances  the  vapor-pressure  of  the  acetone 
is  141  mm.  and  that  of  the  chloroform  is  113  mm.  Assuming  that  the 
apparent  deviation  from  Raoult's  law  is  wholly  due  to  the  formation 
of  a  non-volatile  compound  (CH3)2CO.CHC13,  calculate  the  mol-percent 
of  this  compound  and  the  mol-percents  of  uncombined  (CH3)2CO  and 
CHC13,  a,  from  the  vapor-pressures  of  the  acetone;  and  b,  from  those 
of  the  chloroform.  Ans.  (CH3)2CO.CHC13:  a,  18.2,  b,  22.8. 

In  connection  with  the  deviations  from  Raoult's  law  it  should  be 
borne  in  mind  that,  as  the  mol-fraction  of  any  component  approaches 
unity,  so  that  the  solution  becomes  dilute  with  respect  to  the  other 
component,  its  partial  vapor-pressure  always  approaches  that  re- 
quired by  Raoult's  law,  and  the  partial  vapor-pressure  of  the  other 
component  conforms  to  Henry's  law,  however  great  the  deviation 
may  be  when  both  components  are  present  in  large  proportion. 

Vapor-pressure-composition  diagrams  for  solutions  whose  com- 
ponents do  not  conform  to  Raoult's  law  can  be  constructed  only  with 
the  aid  of  experimentally  determined  data.  By  reference  to  such 
diagrams  the  behavior  of  the  solutions  when  submitted  to  distillation 
at  constant  temperature  can  be  predicted. 

Prob.  36.  —  Construction  of  Vapor-Pressure-Composition  Diagrams.  — 

a.  Draw  on  a  large-scale  diagram  vapor-pressure  curves  representing 
the  partial  vapor-pressure  at  35.2°  of  carbon  bisulfide  and  of  acetone  in 
solutions  of  these  components  throughout  the  whole  range  of  composi- 
tion.    Assume  that  Raoult's  law  and  Henry's  law  hold  with  this  pair 
of  components  up  to  5  mol-percent;  and  make  use  of  the  following 
values  in  millimeters  of  the  vapor-pressures  at  35.2°: 

Mol-percent  of  CSj.  o       i      20    40    60    80    99  100 

Vapor-pressure  of  CS2  o  17.8   274  377  425  460    —  518 

Vapor-pressure  of  (CH3)2CO  353    —    289  255  228  187  20.1     o 

b.  On  the  same  diagram  draw  a  curve  representing  the  total  vapor- 
pressures  of  the  solutions.     Draw  on  the  diagram  dotted  lines  showing 
what  the  partial  and  total  vapor-pressures  would  be  if  the  solutions 
behaved  as  perfect  solutions,    c.  Calculate  the  mol-percents  of  CS2  in 
the  vapor  in  equilibrium  with  the  5,  20,  40,  60,  80,  and  95  mol-percent 
liquid  solutions;  and  on  the  same  diagram  plot  these  vapor-compositions 
against  the  total  pressures  and  draw  a  dotted  line  through  the  points. 

43.  Boiling-Point  of  Concentrated  Solutions  in  General  in  Relation 
to  their  Composition.  —  The  boiling-point-composition  diagrams  for 
solutions  whose  vapor-pressures  do  not  conform  to  Raoult's  law  can 
be  based  upon  direct  experimental  determinations  of  the  boiling-point 


VAPOR-PRESSURE  AND  BOILING-POINT 


73 


of  solutions  of  known  composition  and  upon  analyses  of  the  correspond- 
ing distillates.  All  that  can  be  done,  in  the  way  of  generalization,  is 
to  consider  the  different  types  of  curves  to  which  different  pairs  of 
substances  conform.  The  figure  shows  the  three  types  of  curves 
exhibited  by  substances  miscible  in  all  proportions.  In  each  case 
the  solid  curve  shows  the  compositions  (expressed  as  mol-fractions) 
and  corresponding  boiling-points  of  the  liquid  solutions  at  one  atmos- 
phere; and  the  broken  curve  shows  the  composition  of  the  vapor  that  is 
in  equilibrium  with  these  solutions.  Thus  any  point  on  a  broken  curve 
represents  the  composition  of  the  vapor  of  the  liquid  solution  whose 
point  lies  in  the  same  horizontal  line. 


76° 


0.6      0.5      0.4 
Mol-Fraction 


0.3      0.2      0.1       0 


Curve  I  is  the  experimentally  determined  curve  for  solutions  of 
carbon  tetrachloride  (b.  pt.,  76.7°)  and  carbon  bisulfide  (b.  pt., 
46.3°).  Curve  II  is  that  for  solutions  of  acetone  (b.  pt.,  56.2°)  and 
chloroform  (b.  pt.,  61.3°).  Curve  III  is  that  for  solutions  of  acetone 
(b.  pt.,  56.2°)  and  carbon  bisulfide  (b.  pt.,  46.3°). 


74  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

Solutions  of  type  I,  when  subjected  to  fractional  distillation,  behave 
like  perfect  solutions  (which  form  a  special  case  of  this  type),  and 
may  like  them  be  finally  resolved  into  the  pure  components.  The 
behavior  of  solutions  of  types  II  and  III  on  fractional  distillation 
is  shown  in  the  following  problems. 

Fractional  Distillation  of  Solutions.  — 

Prob.  57.  —  a.  Determine  from  curve  III  in  the  figure  the  boiling- 
point  and  composition  of  the  first  portion  of  distillate  obtained  by  dis- 
tillation of  solutions  of  acetone  and  carbon  bisulfide  containing  70  mol- 
percent  of  acetone,  15  mol-percent  of  acetone,  and  35  mol-percent  of 
acetone,  respectively.  Tabulate  the  boiling-point  and  composition  of 
each  solution  and  distillate,  b.  State  in  what  respects  each  distillate 
and  e,ach  residue  differs  from  the  solution  from  which  it  was  obtained. 
c.  If  each  of  the  solutions  were  submitted  repeatedly  to  fractional  dis- 
tillation, what  would  be  the  composition  of  the  products  finally  obtained 
as  distillate  and  as  residue?  d.  If  of  the  70  mol-percent  solution  1000  g. 
were  so  fractionated,  what  weight  of  each  product  would  be  obtained? 
Ans.  d,  508  g.  of  distillate,  492  g.  of  residue..  • 

Prob.  38.  —  By  reference  to  curve  II  in  the  figure  answer  the  same  ques- 
tions for  solutions  of  acetone  and  chloroform  as  are  asked  in  Prob. 
37&j  and  37^,  for  solutions  of  acetone  and  carbon  bisulfide. 


Just  as  the  solid  lines  in  such  temperature-composition  diagrams 
show  the  boiling-points  of  any  liquid  solution,  so  the  broken  lines 
show  the  condensation-point  at  one  atmosphere  of  vapor  of  any 
composition  ;  the  composition  of  the  liquid  which  first  condenses  out 
of  it  being  given  by  the  corresponding  point  on  the  solid  line.  Such 
diagrams  therefore  serve  to  predict  the  behavior  of  vapors  when 
subjected  to  fractional  condensation,  as  illustrated  by  the  following 
problem. 

The  processes  used  in  chemical  practice  for  the  separation  of 
volatile  liquids,  such  as  alcohol  and  water  or  benzene  and  toluene, 
involve  fractional  distillations  and  condensations  taking  place  in 
accordance  with  the  principles  here  considered. 

Prob.  39.  —  Fractional  Condensation  of  Vapors.  —  a.  A  vapor  com- 
posed of  equimolal  quantities  of  carbon  tetrachloride  and  carbon 
bisulfide  is  cooled  at  i  atm.  till  condensation  begins.  By  referring  to 
curve  I  in  the  figure  find  the  temperature  at  which  condensation  begins, 
and  the  composition  of  the  condensate.  b.  The  vapor  is  gradually 
cooled,  removing  the  condensate  as  it  forms,  till  the  temperature  falls 
to  60°.  Find  the  composition  of  the  condensate  which  is  now  separat- 
ing, and  that  of  the  residual  vapor,  c.  Tabulate  the  composition 
(50  mol-percent)  of  the  original  vapor,  the  average  composition  of 


VAPOR-PRESSURE  AND  BOILING-POINT  75 

the  condensate  obtained  from  it  in  b,  and  the  composition  of  the  residual 
vapor.  Include  in  the  table  also  the  composition  of  the  liquid  which 
upon  distillation  would  furnish  the  original  vapor. 


76  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

V.   DISTRIBUTION  BETWEEN    PARTIALLY   MISCIBLE   SOLVENTS 

44.  Determination  of  Equilibrium-Conditions  by  the    Perpetual- 
Motion  Principle.  —  Before  considering   other    molal    properties   of 
solutions  it  is  desirable  to  present  an  energy  principle  which  is  of  great 
value  in  predicting  the  equilibrium  conditions  of  systems  consisting 
of  two  or  more  phases.    This  principle  is  illustrated  by  Prob.  40  and 
discussed  in  the  succeeding  paragraph. 

Prob.  40.  —  Condition  of  Equilibrium  between  Two  Phases.  —  A 
volatile  substance  S  is  dissolved  in  each  of  two  non-miscible  solvents, 
the  two  solutions  A  and  B  being  shaken  to- 
gether at  some  definite  temperature  till  equi- 
librium is  reached.  They  are  placed,  as  in  the 
figure,  in  contact  with  the  vapor-phase  contain- 
ing the  substance  S  at  a  pressure  equal  to  its 
partial  vapor-pressure  in  solution  A.  Show  that 
if  this  pressure  were  greater  or  less  than  the 
partial  vapor-pressure  of  S  in  the  solution  B,  the 
substance  S  would  pass  continuously  through  the  three  phases  of  the 
system  under  a  difference  of  pressure,  whereby  work  could  be  produced. 

Perpetual  motion  (of  the  kind  here  considered)  signifies  in  general  an 
ideal  process  by  which  an  unlimited  amount  of  work  might  be  pro- 
duced by  a  system  operating  in  surroundings  of  constant  temperature 
and  drawing  from  them  no  work.  Thus,  in  the  case  considered  in 
Prob.  40,  if  the  vapor-pressures  of  the  substance  in  the  two  solutions 
were  different,  a  current  of  its  vapor  would  flow  continuously  from 
one  surface  to  the  other,  and  work  could  be  obtained  from  the  moving 
vapor  for  an  unlimited  period  of  time,  for  example,  by  placing  a  wind- 
mill in  the  vapor-space.  Even  though  a  quantity  of  heat  equivalent 
to  the  work  produced  were  taken  up  from  the  surroundings,  the  process 
would  still  be  a  kind  of  perpetual  motion  which  is  impossible,  as  will 
be  seen  later  in  the  discussion  of  the  second  law  of  thermodynamics. 

The  principle  that  perpetual  motion  of  this  kind  is  impossible  is 
often  employed,  as  in  this  instance,  for  determining  the  conditions  of 
equilibrium  between  the  different  phases  of  a  system.  It  leads  in 
such  cases  to  the  general  conclusion  that,  if  two  phases  are  each  in 
equilibrium  with  a  third  phase,  they  must  be  in  equilibrium  with 
each  other,  with  respect  to  every  substance  that  is  present.  This 
principle  will  hereafter  be  called  simply  the  perpetual-motion  principle. 

45.  Distribution  of  a  Solute  between  Two  Non-Miscible  Solvents. — 
There  has  been  experimentally  established  another  important  law 


DISTRIBUTION  BETWEEN  SOLVENTS  77 

of  perfect  solutions,  as  follows.  At  any  definite  temperature  the 
ratio  between  the  equilibrium  concentrations  CA  and  CB  of  a  chemical 
substance  S  in  two  non-miscible  solvents  (A  and  B)  is  constant, 
whatever  be  the  initial  concentrations^  That  is 

cA/cB=K, 

where  K  is  a  constant,  called  the  distribution-ratio,  determined  by  the 
nature  of  the  substances  A,  B,  and  S,  and  by  the  temperature. 

This  law  may  be  generalized  so  as  to  be  applicable  to  the  distribu- 
tion of  a  definite  chemical  substance  between  any  two  kinds  of  phases. 
Thus  in  its  general  form  it  includes  Henry's  law,  since  the  pressure 
of  a  gas  at  any  definite  temperature  is  proportional  to  its  concentra- 
tion; that  is,  since  p  =  (N/v)RT  =  cRT.  It  is  called  the  law  of  distri- 
bution between  phases,  or  simply  the  distribution  law. 

This  distribution  law,  like  Raoult's  law  and  Henry's  law,  is  a  limit- 
ing law  which  becomes  more  exact  as  the  concentrations  approach  zero, 
but  which  holds  true  approximately  up  to  moderate  concentrations 
such  as  i  molal. 

The  distribution-ratio  of  a  solute  between  water  and  another  solvent 
is  decreased  by  the  addition  of  a  salt  to  the  water  in  accordance  with 
the  principles  of  the  salting-out  effect  stated  in  Art.  39. 

The  distribution  law  between  liquid  phases  can  be  derived  from 
Henry's  law  and  the  perpetual-motion  principle  (as  in  Prob.  41)  for  a 
volatile  solute.  This  derivation  is  general,  since  every  solute  has  a 
vapor-pressure,  even  though  it  may  be  too  small  to  measure. 

In  considering  the  distribution  of  solutes  between  liquid  phases, 
the  concentration  is  usually  expressed  in  mols  per  1000  ccm.  of  solvent, 
instead  of  in  mols  per  1000  grams  of  solvent;  and  the  value  of  the 
distribution-ratio  varies  correspondingly.  In  the  problems  of  this 
article  such  volume  concentrations  are  employed. 

Prob  41.  —Derivation  of  the  Distribution  Law.  — Derive  the  law  of 
the  distribution  of  a  volatile  solute  between  two  solvents  from  Henry's 
law  and  the  conclusion  reached  in  Prob.  40. 

Prob.  42.  —Evaluation  of  the  Distribution-Ratio  from  Vapor-Pressure 
Data.  —  At  25°  the  vapor-pressure  of  ammonia  above  a  o.i  volume- 
molal  (vm.)  solution  of  it  in  chloroform  is  33.25  mm.,  and  above  a  0.5 
vm.  solution  of  it  in  water  is  6.65  mm.  a.  What  is  the  distribution- 
ratio  of  ammonia  between  water  and  chloroform?  b.  What  would  be 
its  distribution-ratio  between  a  0.5  n.  NaCl  solution  and  chloroform? 

Prob.  43. — Extraction  of  Solutes  from  Aqueous  Solutions  by  Organic 
Solvents.  —  The  distribution-ratio  of  an  organic  acid  between  water  and 


78  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

ether  at  20°  is  0.4.  A  solution  of  5  g.  of  the  acid  in  100  ccm.  of  water  is 
shaken  successively  with  three  2o-ccm.  portions  of  ether,  a.  How  much 
acid  is  left  in  the  water?  b.  How  much  acid  would  have  been  left  in  the 
water  if  the  solution  had  been  shaken  once  with  a  6o-ccm.  portion  of 
ether?  Ans.  a,  1.48;  b,  2.00. 

Prob.  44.  —  Determination  of  Complex  Salt  Formation.  —  An  aqueous 
solution  containing  0.25  KC1  and  0.20  HgCl2  in  one  liter  of  water  is 
shaken  with  an  equal  volume  of  benzene  at  25°.  The  benzene  phase  is 
found  by  analysis  to  contain  0.0057  m°l  of  HgCl2  per  liter,  but  no  KC1. 
The  distribution-ratio  of  HgCl2  between  water  and  benzene  at  25° 
is  13.3.  a.  Calculate  the  total  concentration  of  mercuric  chloride  in  the 
aqueous  phase  and  the  concentration  of  the  part  of  it  which  is  combined 
with  the  potassium  chloride  (neglecting  the  salting-out  effect),  b.  The 
complex  salt  has  been  shown  by  other  measurements  to  be  KHgCk 
Tabulate  its  concentration  and  the  concentrations  of  the  (uncom- 
bined)  KC1  and  HgCl2.  Ans.  0.1185  KHgQ3  per  liter. 

*46.  The  Lowering  by  Solutes  of  the  Solubility  of  One  Solvent 
in  Another  Solvent.  —  The  solubility  of  one  solvent,  such  as  water, 
in  another  solvent,  such  as  ether,  is  found  to  be  lowered  by  dissolving 
in  the  first  solvent  a  solute,  such  as  sodium  chloride.  The  general 
law  expressing  this  effect  corresponds  in  form  to  Raoult's  law 
of  vapor-pressure  lowering.  This  law,  called  the  law  of  solubility 
lowering,  may  be  stated  as  follows.  If  in  a  solvent  A,  which  has  a 
small  solubility  SQ  in  another  solvent  B,  there  is  dissolved  a  solute  S 
(not  soluble  in  the  solvent  B)  whose  mol-fraction  in  A  has  any  small 
value  x,  the  solubility  s  of  A  in  B  is  now  less  than  its  original  value 
SQ  to  the  extent  required  by  the  law  that  the  fractional  lowering  of 
solubility  is  equal  to  the  mol-fraction  of  the  solute.  That  is: 

SQ-S  _ 

—  x. 
So 

The  law  of  solubility  lowering  can  be  derived  from  Raoult's  law, 
Henry's  law,  and  the  perpetual  motion  principle,  as  shown  in  the 
following  problem. 

Prob.  45.  —  Derivation  of  the  Law  of  Solubility  Lowering.  —  Show 
how  the  law  of  solubility  lowering  can  be  derived  from  Raoult's  law 
of  vapor-pressure  lowering  with  the  aid  of  the  perpetual-motion  princi- 
ple (applied  to  the  vapor  of  the  substance  A  above  the  two  liquid 
phases)  and  of  Henry's  law  (applied  to  the  equilibrium  of  the  substance 
A  between  the  vapor  phase  and  the  solution  of  A  in  B). 

Prob.  46. — Application  of  the  Law  of  Solubility  Lowering. — The 
solubility  of  isoamyl  alcohol  (C5HnOH)  in  water  at  25°  is  0.280  mol 
per  1000  g.  Calculate  its  solubility  in  water  when  0.5  mol  of  chloro- 
form (CHCls)  is  dissolved  in  1000  g.  of  isoamyl  alcohol. 


DISTRIBUTION  BETWEEN  SOLVENTS  79 

47.   The  Vapor-Pressure  Relations  of  Partially  Miscible  Liquids.  — 

Raoult's  law,  Henry's  law,  and  the  perpetual-motion  principle  make  it 
possible  to  predict  in  part  the  vapor-pressure  relations  of  liquids 
which  have  limited  solubilities  in  each  other,  as  is  illustrated  inProb.47. 

Pr ob.  47.  —  The  Vapor-Pressure-Composition  Diagram  for  Partially 
Miscible  Liquids.  —  When  isoamyl  alcohol  (C6HnOH)  and  water  are 
shaken  together  at  25°  till  equilibrium  is  attained,  two  liquid  phases 
result,  one  containing  36.84  mol-percent,  and  the  other  99.50  mol- 
percent  of  water,  a.  Calculate  the  vapor-pressure  of  the  water  in  each 
of  the  phases  from  the  vapor-pressure,  23.8  mm.,  of  pure  water  at  25°, 
regarding  the  isoamyl  alcohol  present  in  small  proportion  in  the  aqueous 
phase  as  a  perfect  solute,  b.  Construct  a  diagram  showing  the  vapor- 
pressure  of  water  in  all  mixtures  of  these  two  substances  at  25°,  with 
the  aid  of  the  above  values  and  the  following  data.  The  vapor-pres- 
sure of  water  at  25°  is  19.1  mm.  in  a  20.5  mol-percent,  14.5  mm.  in  13.1 
mol-percent,  and  8.4  mm.  in  a  6.9  mol-percent  solution  of  it  in  isoamyl 
alcohol.  Note  that  the  water  must  conform  to  Henry's  law  so  long  as  it 
is  present  at  small  concentration. 

Prob.  48.  —  Steam  Distillation  of  Liquids  Partially  Miscible  with 
Water.  —  Aniline  (C6H5NH2)  is  distilled  with  steam  at  i  atm.  The 
vapor-pressure  of  aniline  is  46  mm.  at  100°  and  40  mm.  at  97°.  Water 
and  aniline  have  limited  solubilities  in  each  other;  one  phase  containing 
at  100°  1.5  mol-percent  of  aniline,  and  the  other  68  mol-percent  of 
aniline.  Find  the  boiling-point  of  the  mixture,  and  the  number  of  grams 
of  aniline  distilling  over  with  each  gram  of  water.  Assume  that  the 
solubilities  do  not  change  appreciably  within  the  small  temperature 
interval,  and  that  the  vapor-pressure  of  the  aniline  in  the  68  mol- 
percent  solution  is  lowered  only  half  as  much  as  Raoult's  law  requires. 
Ans.  0.26  g. 


8o  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

VI.     FREEZING-POINT  OF   SOLUTIONS 

48.  Freezing-Point  and  Its  Relation  to  Vapor-Pressure  and  Molal 
Composition.  —  The  freezing-point  of  a  liquid  is  that  temperature 
at  which  the  solid  solvent  and  the  liquid  coexist  in  equilibrium  with 
each  other.  The  solid  which  separates  from  the  solution  commonly 
consists  of  the  pure  solvent;  and  this  is  assumed  to  be  the  case  through- 
out the  following  considerations.  Jn  a.ny  such  case  the  freezing-point 
of  a  solvent  is  lowered  by  dissolving  another  substance  in  it. 

At  the  freezing-point  the  vapor-pressure  of  the  solvent  in  the  solu- 
tion and  of  the  pure  solid  solvent  must  be  equal  (as  shown  in  Prob. 
49);  and  accordingly,  from  the  vapor-pressure-temperature  curves 
of  the  liquid  solvent  and  solid  solvent  the  freezing-points  of  solutions 
can  be  derived  from  their  vapor-pressures  (as  in  Prob.  50). 

Prob.  4Q.  —  Equality  of  the  Vapor-Pressures  at  the  Freezing-Point.  — 
Prove  that  at  the  freezing-point  of  a  solution  its  vapor-pressure  and 
that  of  the  solid  which  separates  from  it  must  be  equal,  by  showing 
that  otherwise  perpetual  motion  would  result. 

Prob.  50.  —  Evaluation  of  Freezing-Points  from  Vapor-Pressure  Data.  — 
The  vapor-pressures  of  ice  and  (supercooled)  water  are  as  follows: 

o°         -i°       -2°       -3°       -4° 

Water    .     .     .     4.579     4.255     3.952     3.669     3.404  mm. 
Ice    .     .     .     .     4.579     4.215     3.879     3.566     3.277  mm. 

a.  Calculate  by  Raoult's  law  the  vapor-pressure  at  each  of  these 
temperatures  of  a  solution  consisting  of  3  mols  of  solute  and  97  mols  of 
solvent.     Plot  on  a  diagram  the  vapor-pressures  of  this  solution,  of 
water,  and  of  ice  as  ordinates  against  the  temperatures  as  abscissas, 
using  a  scale  large  enough  to  enable  o.ooi  mm.  to  be  estimated  (by 
including  on  th£  plot  pressures  ranging  only  from  3.20  to  4.60  mm.). 

b.  Determine  from  the  plot  the  freezing-point  of  the  solution. 

The  general  relations  between  freezing-point  and  vapor-pressure 
are  illustrated  by  the  diagram  on  the  following  page. 

A  consideration  (as  in  ProJ),  51)  of  the  geometrical  relations  of  the 
diagram /shows  tha^t  the  vapor-pressure  lowering  po—p  caused  by  a 
solute  is  for  a  dilute  solution  related  to  the  freezing-point  lowering 
To—  T  in  the  way  expressed  by  the  equation: 

po—p      dpi  _  dpi  ,v 

r0-r    dT    dT 

In  this  expression  the  two  derivatives  represent  the  rate  of  change 
with  the  temperature  of  the  vapor-pressure  of  the  solid  solvent  (ice) 
and  of  the  liquid  solvent,  respectively. 


FREEZING-POINT 


81 


Prob.  51.  —  Derivation  of  the  Relation  between  Freezing- Point  and 
Vapor-Pressure.  —  From  the  geometrical  relations  of  the  following 
diagram  derive  equation  (i)  given  in  the  text. 


o         T          TO 

Freezing-Point 

From  the  approximate  Clapeyron  equation  (Art.  33)  expressions 
for  the  temperature-coefficients  of  the  vapor-pressures  of  the  solid 
solvent  and  liquid  solvent  can  be  obtained  in  terms  of  the  molal  heats 
of  vaporization  A#s  and  A#L  of  the  solid  solvent'  and  the  liquid 
solvent,  respectively.  Since  the  difference  between  these  two  heats 
of  vaporization  can  be  shown  by  the  law  of  initial  and  final  states 
(Art.  23)  to  be  equal  to  the  fcnolal  heat  of  fusion  AZTojbf  the  solid 
solvent,  the  following  expressions,  identical  in  form  with  those  for  the 
boiling-point  raising  (Art.  37),  are  readily  obtained  (as  in  Prob.  52) 
for  the  freezing-point  lowering  of  a  perfect  solution:  . 

RT0*    pQ-p     RTJ 


In  these  equations  T0  is  the  freezing-point  of  the  solvent,  T  that  of  a 
solution  containing  the  solute  at  the  mol-fraction  x,  and  A#o  is  the 
heat  absorbed  by  the  fusion  of  one  mol  of  the  solvent  at  TV 

Prob.  52. — Derivation  of  the  Freezing-Point  Equations  for  Perfect 
Solutions.  —  Derive  equation  (2)  from  equation  (i),  indicating  the 
principle  involved  in  each  step  of  the  process,  and  proving  that  the 
heat  of  fusion  is  equal  to  the  difference  in  the  two  heats  of  vaporization. 

The  quantity  /JTV/A&o  occurring  in  equation  (2)  is  evidently  a 
constant  characteristic  of  the  solvent.  Representing  this  quantity, 
which  may  be  called  the  freezing-point  constant,  by  a  single  letter  F0, 
and  replacing  the  mol-fraction  x  by  the  mol-ratio  N/^Vo,  which  becomes 


82  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

identical  with  it  as  its  value  approaches  zero,  the  law  of  freezing-point 
lowering  for  perfect  solutions  may  be  expressed  by  the  equation: 


Instead  of  this  freezing-point  constant,  the  related  mold  freezing- 
point  lowering,  denned  in  analogy  with  the  molal  boiling-point  raising 
(Art.  37)  to  be  the  freezing-point  lowering  per  mol  of  solute  in 
1000  grams  of  solvent,  is  commonly  recorded  in  chemical  literature. 

Pr  ob.  53.  —  Evaluation  of  the  Freezing-Point  Constants.  —  When  one 
gram  of  ice  at  o°  melts,  the  heat  absorbed  is  79.7  cal.  a.  What  is  the 
freezing-point  constant  for  water?  b.  What  is  its  molal  freezing-point 
lowering?  Ans.  6,1.858  (1.86). 

*The  preceding  equations  are  rigorously  exact  only  when  the  lower- 
ings  of  the  freezing-point  and  of  the  vapor-pressure  of  the  solvent  are 
infinitesimal;  for  in  the  derivation  of  equation  (i)  all  the  vapor- 
pressure  curves  were  regarded  as  straight  lines,  and  those  for  the 
solvent  and  solution  were  regarded  as  parallel.  A  consideration  of 
the  figure  at  the  beginning  of  this  article  shows,  however,  that  the- 
following  differential  equation,  corresponding  to  equation  (i)  above, 
is  exact  for  any  solution  whatever  containing  the  solute  at  any  mol- 
fraction  x  whatever: 


*By  substituting  for  the  first  of  these  derivatives  the  expression 
given  by  the  corrected  Raoult  equation  (Art.  36),  and  for  the  last  two 
derivatives  the  expressions  given  by  the  exact  Clapeyron  equation 
(Art.  33),  neglecting  only  the  volume  of  the  solid  or  liquid  in  compari- 
son with  that  of  the  vapor,  there  is  obtained  the  following  equation  : 


In  this  equation  —dT  denotes  the  lowering  produced  in  the  freezing- 
point  T  of  a  solution  by  increasing  the  mol-fraction  x  of  the  solute  by 
dx,  and  A#  denotes  the  heat  absorbed  by  the  fusion  of  one  mol  of  the 
solid  solvent  into  an  infinite  volume  of  the  solution  at  its  freezing- 
point  T. 

*Prob.  54.  —  Freezing-Point  of  Solutions  Conforming  to  Raoult's  Law 
at  Any  Concentration.  —  Derive  equation  (5)  from  equation  (4)  by 
the  method  indicated  in  the  text. 


FREEZING-POINT  83 

This  equation  will  be  seen  to  correspond  in  form  to  equation  (5.) 
of  Art  37  for  the  boiling-point  raising.  The  equation  is  exact  (except 
for  the  error  caused  by  neglecting  the  volume  of  the  solid  or  liquid 
in  the  Clapeyron  equation)  at  any  concentration  up  to  which  the 
corrected  Raoult  equation  holds  true.  It  may  be  integrated,  usually 
with  negligible  error,  under  the  assumption  that,  within  the  limits  of 
temperature  and  concentration  involved,  AH  does  not  vary  with 
the  temperature  or  with  the  mol-fraction  of  the  solute,  and  that  it 
has  the  same  value  as  the  heat  of  fusion  A#0  of  the  pure  solvent  at  its 
freezing-point  TV 

*The  freezing-points  of  solutions  that  do  not  conform  to  Raoult's 
law  cannot  be  calculated  from  the  molal  composition.  The  relation 
between  freezing-point  and  molal  composition  can,  of  course,  be 
experimentally  determined;  and  the  results  may  be  represented  by 
freezing-point-composition  diagrams,  analogous  to  the  boiling-point- 
composition  diagrams  of  Art.  43.  Such  diagrams  are  considered  in 
Chapter  VIII. 

49.  Determination  of  Molecular  Weights.  —  With  the  aid  of  the 
laws  of  freezing-point  lowering  stated  in  Art.  48  the  molecular  weigtfts 
of  substances  dissolved  in  various  solvents  can  be  calculated. 

Prob.  55.  —  Variation  of  Molecular  Weight  with  the  Nature  of  the 
Solvent.  —  A  solution  of  0.60  g.  of  acetic  acid  in  50.0  g.  of  water  freezes 
at  —0.376°.  A  solution  of  2.32  g.  of  acetic  acid  in  100  g.  of  benzene 
freezes  0.970°  lower  than  pure  benzene.  The  freezing-point  constant 
for  benzene  is  65.4.  Calculate  the  molecular  weight  of  acetic  acid  in 
each  of  these  solvents,  and  state  what  the  results  show  in  regard  to  its 
molecular  formula  in  each  solvent. 

Determinations  of  the  molecular  weights  of  dissolved  substances 
by  the  freezing-point  method  have  confirmed  the  conclusions  derived 
from  boiling-point  determinations,  and  stated  in  Art.  38,  as  to  the 
dissociating  effect  of  various  solvents. 


84  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

VII.    OSMOTIC  PRESSURE   OF   SOLUTIONS 

50.  Osmotic  Pressure.  —  When  a  solution  S  is  separated  from  a 
pure  solvent  W,  as  illustrated  in  the  figure,  by  a  wall  aa  which  allows 

this  solvent  to  pass  through  it,  but  prevents 
entirely  the  passage  of  the  solute,  the  solvent 
is  drawn  through  the  wall  into  the  solution. 
This  flow  of  solvent  may,  however,  be  pre- 
vented by  exerting  a  pressure  fa  on  the 
solution  greater  by  a  definite  amount  than 
the  pressure  pi  upon  the  pure  solvent;  and 
the  solvent  may  be  forced  out  of  the  solution 
by  exerting  a  still  greater  pressure  upon  the 

solution.  The  difference  of  pressures  on  solution  and  solvent  which 
produces  a  condition  of  equilibrium  such  that  there  is  no  tendency  of 
the  solvent  to  flow  in  either  direction  is  called  the  osmotic  pressure  P  of 
the  solution. 

Walls  of  the  kind  just  described  are  known  as  semipermeable  walls. 
Certain  animal  membranes,  such  as  parchment  or  bladder,  are  per- 
meable for  water,  but  no^|r  certain  solutes  of  high  molecular  weight 
(the  so-called  colloids).  The  walls  of  some  animal  and  plant  cells 
are  very  perfect  semipermeable  walls.  The  most  satisfactory  artificial 
semipermeable  walls  have  been  made  by  precipitating  copper  ferro- 
cyanide  within  the  pcres  of  an  unglazed  porcelain  cell,  which  gives 
the  precipitate  sufficient  rigidity  to  withstand  high  osmotic  pressures. 
The  cell,  filled  with  the  solution  and  immersed  in  pure  water,  is  con- 
nected with  a  manometer  whose  mercury  column  is  in  direct  contact 
with  the  solution.  By  means  of  such  cells  exact  measurements  of 
the  osmotic  pressure  of  aqueous  solutions  of  cane-sugar  and  glucose 
have  been  made  up  to  pressures  of  230  atmospheres. 

Osmotic  pressure  plays  a  very  important  part  in  the  physiological 
processes  taking  place  in  the  bodies  of  animals  and  plants.  In  the 
study  of  the  general  principles  of  chemistry,  it  is  of  value  because  it 
is  a  property  which,  like  vapor-pressure,  enables  the  various  properties 
of  solutions  to  be  correlated  and  their  energy  relations  to  be  treated 
upon  the  basis  of  one  simple  fundamental  concept.  The  correspond- 
ence shown  in  Art.  5  2  to  exist  between  the  laws  of  the  osmotic  pressure 
of  dilute  solutions  and  those  of  the  pressure  of  perfect  gases  makes 
possible,  moreover,  a  closely  analogous  treatment  of  these  two 
states. 


OSMOTIC  PRESSURE  85 

*51.  Relation  of  Osmotic  Pressure  to  Vapor-Pressure.  —  In  the 

osmotic  arrangement  represented  in  the  figure  of  Art.  50,  when  the 
pressures  pi  and  p2  are  such  that  there  is  equilibrium  and  therefore  no 
tendency  for  the  solvent  to  pass  through  the  semipermeable  wall,  it 
follows  from  the  perpetual  motion  principle  that  the  vapor-pressure  of 
the  solvent  in  the  pure  state  must  be  equal  to  its  vapor-pressure  in  the 
solution.  For,  if  a  vapor  phase  were  in  contact  (through  walls  per- 
meable only  for  the  vapor)  both  with  the  solvent  and  the  solution,  there 
would  evidently  be  a  continuous  flow  of  the  solvent-substance  through 
the  vapor-phase  and  back  through  the  wall  between  the  two  liquid 
phases,  unless  its  vapor-pressures  in  these  liquid  phases  were  equal. 
That  this  condition  is  realized  results  from  the  fact  that  the  vapor- 
pressure  of  a  liquid  is  increased  by  increasing  the  pressure  upon  it. 
Thus  the  larger  pressure  p%  on  the  solution  increases  its  vapor-pressure 
up  to  the  value  of  the  vapor-pressure  of  the  solvent  under  the  smaller 
pressure  pi. 

The  effect  of  pressure  on  vapor-pressure  may  be  now  considered 
from  a  quantitative  standpoint.  It  can  be  shown  (as  in  Prob.  56) 
that  the  increase  dp  in  the  vapor-pressure  p  of  a  liquid  caused  by  an 
increase  dp^  in  the  pressure  p^  on  the  liquid  is  expressed  by  the  follow- 
ing equation,  in  which  z>L  represents  the  volume  of  any  definite  weight 
of  the  liquid  at  the  pressure  £L,  and  v  represents  the  volume  of  the 
same  weight  of  the  vapor  at  the  pressure  p: 

(i) 

In  case  the  vapor  may  be  regarded  as  a  perfect  gas,  this  equation 
takes  the  following  form,  in  which  z>L  represents  the  volume  of  one 
mol  of  the  liquid  at  the  pressure  p^: 


By  integration  of  this  equation  under  the  assumption  that  the 
volume  of  the  liquid.  VL  does  not  change  with  the  pressure  there 
results  the  following  expression,  in  which  p  and  p0  denote  the  vapor- 
pressures  of  the  solvent  when  the  pressures  on  it  are  pL  and  zero, 
respectively: 


_ 

/>0  "  RT 


86 


THE  MOLAL  PROPERTIES  OF  SOLUTIONS 


£?ELJ& 


Effect  of  Pressure  on  the  Vapor-Pressure  of  Liquids.  — 

Prob.  56.  —  Consider,  as  illustrated  in  the  figure,  a  column  of  a  pure 
liquid  contained  in  a  porous  tube  impermeable  to  it,  but  permeable  to 
its  vapor;  and  consider  this  tube  to  be  surrounded  by  the  vapor  of  the 
liquid;  the  whole  system  being  at  a  constant  tempera- 
ture T.  When  there  is  equilibrium  the  vapor-pressure 
of  the  liquid  at  any  level  must  evidently  be  equal  to 
the  pressure  of  the  vapor  at  that  level.  The  pressure 
of  the  liquid  or  of  the  vapor  must,  however,  be  greater 
at  a  lower  than  a  higher  level  by  the  (hydrostatic) 
pressure  of  the  intervening  column  of  liquid  or  vapor. 
a.  Formulate  an  expression  for  the  increase  dpi.  of  the 
pressure  pi.  of  the  liquid,  and  one  for  the  increase  dp  of 
the  pressure  p  of  the  vapor,  corresponding  to  an  in- 
crease dl  in  the  distance  /  beneath  the  surface,  b.  By 
combining  these  expressions,  derive  equation  (i)  given 
in  the  text.  c.  Integrate  this  equation,  assuming  that 
the  vapor  conforms  to  the  perfect-gas  laws  and  that 
the  volume  of  the  liquid  does  not  vary  with  the  pres- 
sure upon  it,  so  as  to  obtain  a  relation  between  the 
vapor-pressures  po  and  p  of  the  liquid  at  the  top  and 
bottom  of  the  column  in  terms  of  its  hydrostatic  pressure  pi,. 

Prob.  57.  —  Find  the  ratio  of  the  vapor-pressure  of  water  at  4°  and 
10  atm.  to  that  at  4°  and  the  pressure  (6  mm.)  of  the  saturated  vapor. 

It  is  evident  now  that,  by  applying  a  sufficient  negative  pressure 
(or  suction)  to  a  liquid  solvent,  its  vapor-pressure  can  be  reduced 
from  its  normal  value  pQ  to  the  value  p  which  it  normally  has  in  any 
definite  solution,  and  that  then  it  would  be  in  equilibrium  with  that 
solution  if  placed  in  communication  with  it  through  a  semipermeable 
wall.  (Liquids  can  in  fact  with  suitable  precautions  be  subjected 
to  negative  pressures  of  many  atmospheres,  without  the  column 
breaking.)  The  difference  in  the  pressures  on  the  solution  and  the 
solvent,  defined  to  be  the  osmotic  pressure  P,  is  evidently  numerically 
equal  to  this  pressure  />L  applied  to  the  solvent,  but  opposite  in  sign ; 
that  is,  P=  —  pi..  Hence  it  follows  from  equation  (3)  that  the  osmotic 
pressure  at  the  temperature  T  of  a  solution  in  a  solvent  whose  vapor- 
pressure  is  p  in  the  solution  and  p0  in  the  pure  state,  and  whose  molal 
volume  is  VL,  is  given  by  the  expression: 

This  expression  is  exact  even  at  large  concentrations  provided  the 
vapor  conforms  to  the  perfect-gas  laws,  except  for  the  usually  negli- 
gible effect  of  pressure  on  the  volume  z>L  of  the  solvent. 


OSMOTIC  PRESSURE  87 

Relation  of  Osmotic  Pressure  to  Vapor-Pressure.  — 

Prob.  58.  —  a.  How  much  must  the  pressure  on  water  be  reduced 
in  order  that  it  may  be  in  equilibrium  through  a  semipermeable  wall 
with  an  aqueous  solution  which  has  at  25°  a  vapor-pressure  98%  as 
great  as  that  of  water?  b.  What  is  the  osmotic  pressure  of  this 
solution  at  25°? 

Prob.  59.  —  At  100°  the  vapor-pressure  of  a  solution  consisting  of 
28  g.  of  NaCl  and  100  g.  of  water  is  624  mm.  What  is  its  osmotic 
pressure?  The  specific  volume  of  water  at  100°  is  1.043.  Ans.  320  atm. 

When  the  vapor  does  not  conform  to  the  perfect-gas  laws  an  exact 
expression  may  be  readily  obtained  by  substituting  for  v  in  equation 
(i)  the  expression  RT(i  +  a  p)/p  given  by  the  equation  of  state  for 
gases  at  moderate  pressure  (Art.  15),  and  integrating  as  before.  This 
leads  to  the  result  that  the  osmotic  pressure  equation  may  be  corrected 
for  deviation  of  the  vapor  from  the  perfect-gas  laws  by  writing  it  in 
the  form: 

P=^(log>-°+a(^)).  (5) 

52.   Relation  between  Osmotic  Pressure  and  Molal  Composition.  — 

The  osmotic  pressure  P  of  concentrated  solutions  whose  vapor-pres- 
sure conforms  to  Raoult's  law  is  expressed  by  the  following  equation, 
which  may  readily  be  derived  (as  shown  in  Prob.  60)  from  equation 
(4)  or  (5)  of  Art.  51. 

(i) 


In  this  equation  £L  represents  the  molal  volume  of  the  solvent,  and 
N  denotes  the  number  of  mols  of  solute  associated  with  No  mols  of 
solvent. 

For  dilute  solutions,  for  which  N/  N0  is  small,  this  equation  can  be 
simplified  (as  shown  in  Prob.  61)  by  expanding  the  logarithm  into  a 
series,  neglecting  all  the  terms  except  the  first  one,  and  transforming. 
The  following  expression  then  results: 

Pv0=NRT.  (2) 

In  this  expression  P  is  the  osmotic  pressure  at  the  temperature  T  of  a 
solution  consisting  of  N  mols  of  solute  and  NQ  mols  of  solvent  whose 
volume  in  the  pure  state  is  i>0,  and  R  is  the  gas-constant. 

This  equation  is  seen  not  only  to  be  identical  in  form  with  the 
perfect-gas  equation,  but  to  contain  the  same  constant  R.  This 
shows  that  a  substance  in  dilute  solution  produces  an  osmotic  pressure 
which  is  equal  to  the  pressure  it  exerts  at  the  same  temperature  as  a 
perfect  gas  when  present  at  the  same  volume  concentration. 


88  THE  MOLAL  PROPERTIES  OF  SOLUTIONS 

The  osmotic  pressure  of  concentrated  solutions  to  which  Raoult's 
law  does  not  apply  cannot  be  calculated  from  the  composition  of  the 
solution.  It  may,  however,  be  determined  not  only  by  direct  measure- 
ment, but  also  by  calculation  from  vapor-pressure  measurements,  by 
equations  (4)  and  (5)  of  Art.  51. 

Derivation  of  the  Osmotic-Pressure  Equations.  — 

*Prob.  60.  —  Derive  equation  (i)  of  this  article,  a,  from  equation  (4), 
and  b,  from  equation  (5)  of  Art.  51. 

Prob.  61. — Derive  equation  (2)  from  equation  (i)  in  the  way  in- 
dicated in  the  above  text. 

Prob.  62.  —  Osmotic  Pressure  in  Relation  to  Hydrostatic  Pressure.  — 
The  lower  end  of  a  vertical  tube  is  closed  with  a  semipermeable  wall 
and  is  dipped  just  beneath  the  surface  of  pure  water.  A  o.i  molal 
solution  of  cane-sugar  (CiaHaOu)  of  density  1.014  is  poured  into  the 
tube  until  the  hydrostatic  pressure  at  the  semipermeable  wall  is  sufficient 
to  prevent  water  from  entering  the  solution.  The  temperature  is  4°. 
Find  the  height  of  the  column  in  meters.  Ans.  23.6 


REVIEW  OF  THE  PRINCIPLES  89 

VIII.    REVIEW  OF  THE  PRINCIPLES  RELATING  TO  THE  MOLAL 
PROPERTIES   OF  SOLUTIONS 

53.  Review  Problems.  —  The  following  problems  afford  a  review 
and  give  additional  applications  of  the  principles  relating  to  the 
molal  properties  of  solutions. 

Prob.  63.  —  a.  Summarize  the  equations  expressing  for  dilute  solutions 
the  approximate  relations  between  molal  composition  and  (i)  vapor- 
pressure,  (2)  boiling-point,  (3)  freezing-point,  (4)  osmotic  pressure. 
State  explicitly  what  each  symbol  signifies. 

*Prob.  64.  —  Summarize  the  corresponding  equations  which  hold  true 
for  perfect  solutions  of  any  concentration. 

Prob.  65.  —  Calculate  by  the  laws  of  dilute  solutions  the  fractional 
vapor-pressure  lowering,  the  boiling-point  raising,  the  freezing-point 
lowering,  and  the  osmotic  pressure  at  5.5°,  of  a  solution  containing  o.i 
mol  of  a  nonvolatile  solute  in  1000  g.  of  benzene,  with  the  aid  of  the 
following  data.  The  heat  of  fusion  of  one  gram  of  benzene  at  its 
freezing-point,  5.5°,  is  30.2  cal.  The  heat  of  vaporization  of  one  gram 
of  benzene  at  its  boiling-point,  80.  i°,  is  93.0  cal.  The  density  at  the 
freezing-point  is  0.895. 

*Prob.  66.  —  a.  Calculate  by  the  laws  of  concentrated  perfect  solu- 
tions the  fractional  vapor-pressure  lowering,  the  boiling-point  raising, 
the  freezing-point  lowering,  and  the  osmotic  pressure  at  5.5°,  of  a 
solution  containing  2  mols  of  a  non-volatile  solute  in  1000  g.  of  benzene, 
using  the  data  of  Prob.  65,  and  assuming  that  the  vapor  conforms  to 
the  perfect-gas  laws.  b.  Calculate  the  ratio  of  each  of  these  values 
to  the  corresponding  one  for  the  solution  of  Prob.  65.  (Note  that  the 
value  of  this  ratio  would  be  20.0  if  the  equations  for  dilute  solutions 
were  applicable  to  this  concentrated  solution.)  c.  Assuming  the  solute 
to  be  volatile  and  to  have  a  partial  vapor-pressure  of  20  mm.  in  the 
solution  at  its  boiling-point,  calculate  this  boiling-point.  Ans.  c,  84.2°. 

Prob.  67.  —  a.  Carbon  bisulfide  boils  at  46°  at  i  atm.  Its  molal 
heat  of  vaporization  at  this  temperature  is  6430  cal.  What  is  its  boil- 
ing-point constant?  b.  A  solution  of  15.5  g.  of  phosphorus  (at.  wt., 
31.0)  in  1000  g.  of  CS2  boils  0.300°  higher  than  pure  CSa.  What  is  the 
molecular  weight  and  what  is  the  molecular  formula  of  phosphorus  in 
this  solvent? 

-Prob.  68.  — If  127  g.  of  iodine  (at.  wt.,  127)  were  added  to  the  solution 
of  Prob.  676,  how  much  higher  than  the  boiling-point  of  pure  CSa 
would  that  of  the  solution  be:  a,  in  case  the  iodine  remained  uncom- 
bined  in  the  form  of  I2;  &,  in  case  it  all  combined  with  the  phosphorus 
forming  P4I8;  c,  forming  P2I4;  d,  forming  PI2? 

*Prob.  dp. —  Human  blood  freezes  at  -0.56°.  Find  its  osmotic 
pressure  at  37°.  Ans.  7.6  atm. 

-  Prob.  70.  —  At  25°  the  distribution-ratio  of  Br2  between  carbon  tetra- 
chloride  and  water  is  38  expressed  in  volume  concentrations;  and 
the  pressure  of  bromine  above  a  0.05  molal  solution  of  Br2  in  water  is 


90  THE   MOLAL  PROPERTIES  OF  SOLUTIONS 

50  mm.     Assuming  that  one  liter  of  this  solution  is  shaken  with  50  ccm. 
of  carbon  tetrachloride,  calculate  the  pressure  of  the  bromine  over  the 
carbon- tetrachloride  phase.     (Bromine  exists  in  all  three  phases  only 
%  as  Br2.)    Ans.  17.2  mm. 

Prob.  71.  — At  i  atm.  pure  nitric  acid  has  a  boiling-point  of  86°.  A 
solution  of  nitric  acid  and  water  of  the  composition  HN03  + i.6H2O 
distils  at  i  atm.  at  a  constant  temperature  of  121°.  Make  a  diagram 
showing  the  character  of  the  boiling-point-composition  curve  for  this 
pair  of  substances.  Draw  in  on  the  diagram  a  curve  representing  in  a 
general  way  the  composition  of  the  vapor  in  equilibrium  with  any  solu- 
tion at  its  boiling-point.  State  what  products  would  finally  result  as 
distillate  and  residue  from  the  fractionation  of  the  three  solutions, 

HN03+H2O,  HNO3  +  i.6H2O,  HNO3+3H2O. 

Prob.  72.  —  Upon  partial  distillation  at  constant  pressure  a  solution 
of  two  components  A  and  B  containing  20%  of  A,  having  a  boiling- 
point  of  80°,  yields  a  residue  containing  15%  of  A;  and  a  mixture  con- 
taining 75%  of  A,  having  a  boiling-point  of  60°,  yields  a  distillate 
containing  70%  of  A.  Draw  a  diagram  which  will  show  in  a  general 
way  the  character  of  the  liquid-composition  and  of  the  vapor-composi- 
tion curve.  Predict  what  products  would  finally  result  as  distillate 
and  residue  from  the  fractionation  of  the  20%  and  of  the  75%  solution. 


CHAPTER  IV 
THE  ATOMIC  PROPERTIES   OF   SOLID   SUBSTANCES 


I.    THE  HEAT-CAPACITY   OF   SOLID   SUBSTANCES 

54.  Properties  in  the  Solid  State.  —  Many  of  the  properties  of 
substances  in  the  gaseous  and  dissolved  states  are  primarily  deter- 
mined, as  has  been  shown  in  the  preceding  chapters,  by  the  number  of 
molecules  present;  but  in  the  solid  state  such  molal  properties  are  met 
with  only  in  the  case  of  substances  present  at  small  concentration  in 
solid  solutions  (considered  in  Art.  123).     In  the  solid  state  there  are, 
however,   certain   properties,   here  called  atomic   properties,  which 
depend  primarily  either  on  the  number  of  atoms  present  in  the  unit- 
weight  of  the  substance,  or  on  the  number  of  atoms  present  in  each 
molecule  of  the  substance.    The  heat-capacity  of  solid  substances  is  a 
property  of  the  first  of  these  types;  and  isomorphism,  or  the  power 
of  two  substances  of  assuming  nearly  the  same  crystal  structure  and 
of  forming  with  each  other  a  continuous  series  of  solid  solutions,  is  a 
property  of  the  second  of  these  types.     The  heat-capacity  of  solid 
substances,  which  is  of  importance  not  only  in  its  relations  to  the 
atomic  theory,  but  also  in  thermodynamic  considerations,  will  alone 
be  considered  here. 

55.  The  Heat-Capacity  of  Solid  Elementary  Substances.  —  Meas- 
urements have  shown  that  at  room  temperature  all  elementary  sub- 
stances with  atomic  weights  above  35,  and  the  metallic  elementary 
substances  with  smaller  atomic  weights  than  35,  have  values  of  the 
atomic  heat-capacity  (defined  as  in  Art.  25)  not  differing  greatly 
from  each  other.    This  approximate  principle  is  known  as  Dulong 
and  Petit1  s  law. 

The  average  value  of  the  atomic  heat-capacity  at  constant  pressure 
at  20°  for  these  elements  is  6.2  calories  per  degree.  Deviations  of 
+  0.4  unit  are  not  uncommon;  and  deviations  of  +0.5  to  +0.9  unit 
are  exhibited  by  some  elements  (for  example,  by  sodium,  potassium 
and  iodine)  which  at  room  temperature  are  not  much  below  their 
melting-points.  In  the  case  of  the  non-metallic  elements  with  smaller 
atomic  weights  than  35  the  atomic  heat-capacity  has  a  value  much 
smaller  than  6.2  at  room  temperature;  thus  the  value  for  boron  is  2.6, 
for  graphite  1.9,  for  silicon  4.8,  for  phosphorus  5.6,  and  for  sulphur  5.5. 

91 


92  ATOMIC  PROPERTIES  OF  SOLID  SUBSTANCES 

The  corresponding  average  value  of  the  atomic  heat-capacity  at 
constant  volume  at  room  temperature  is  5.9;  and  from  this  simpler, 
more  fundamental  quantity  the  values  for  the  separate  substances 
show  considerably  smaller  deviations. 

It  is  interesting  to  note  that  this  average  atomic  heat-capacity 
is  very  nearly  twice  the  translatory  kinetic  energy  (2.98  cal.)  taken  up 
per  degree  by  the  molecules  of  a  gaseous  or  liquid  substance  (Art.  19). 
The  kinetic  factors  determining  this  roughly  constant  value  for  solid 
elementary  substances,  and  causing  the  marked  variation  of  the  heat- 
capacity  with  the  temperature  to  be  now  mentioned,. are  too  complex 
and  hypothetical  to  be  here  considered. 

The  effect  of  temperature  on  the  atomic  heat-capacity  at  constant 
pressure  is  shown  in  the  figure.  It  will  be  seen  that  at  very  low 
temperatures  (below  200°  A)  the  atomic  heat-capacity  of  solid  elemen- 
tary substances  varies  much  with  the  nature  of  the  substance  and  is 
always  much  smaller  than  at  room  temperature;  that  with  rising 
temperature  it  increases  at  different  rates  in  the  case  of  different 
substances;  and  that  at  a  temperature,  varying  with  the  different 
substances  but  usually  not  far  from  the  room  temperature,  it  attains 
the  average  value,  and  then  as  a  rule  increases  only  slowly  with  further 
increase  of  temperature. 


0         50       100       150      200      250       300      350      400       450      500 
Absolute  Temperature 


HEAT-CAPACITY  OF  SOLID  SUBSTANCES 


93 


56.  The  Heat-Capacity  of  Solid  Compound  Substances.  —  Another 
roughly  approximate  principle,  called  Kopp's  law,  has  been  found 
experimentally  to  express  the  heat-capacity  of  solid  inorganic  com- 
pounds at  room  temperature.  It  has  been  found,  namely,  that  the 
formal  heat-capacity  is  approximately  an  additive  property,  that  is, 
one  whose  value  can  be  approximately  calculated  by  adding  together 
certain  values  representing  the  heat-capacity  of  the  elements  con- 
tained in  the  compound.  This  principle  is  expressed  by  the  following 
equation,  which  shows  at  the  same  time  the  values  of  the  constants 
(the  so-called  atomic  heat-capacities)  for  the  common  elements: 


In  this  equation  Cp  represents  the  formal  heat-capacity  of  the  com- 
pound at  constant  pressure  at  room  temperature;  no,  nu,  nc,  n$,  WB? 
np,  and  nsi  are  the  number  of  atomic  weights  of  oxygen,  hydrogen, 
carbon,  sulfur,  boron,  phosphorus,  and  silicon  present  in  one  formula- 
weight  of  the  compound;  and  HE  is  the  number  of  atomic  weights  of 
any  other  element  so  present.  The  values  given  for  the  constants 
are  average  values  derived  from  heat-capacity  measurements  with 
solid  compounds.  They  do  not  accurately  represent  the  atomic 
heat-capacities  of  the  solid  elementary  substances. 

The  following  table  illustrates  the  degree  of  correspondence  which 
exists  between  the  values  of  the  formal  heat-capacity  so  calculated  and 
those  measured  experimentally.  It  will  be  observed  that  differences 
of  ten  percent  between  the  calculated  and  measured  values  are  not 
uncommon. 

HEAT-CAPACITIES  OF  SOLID  COMPOUND  SUBSTANCES 

Substance 

H20  (ice) 

A120, 

Fe208 

80,8, 

KC1 

PbCl2 

CaCOs 

KClOs 

Prob.  i.  —  Specific  Heat-Capacities  Calculated  by  the  Additivity 
Principle.  —  Calculate  an  approximate  value  at  20°  of  the  specific 
heat-capacity  at  constant  pressure  of:  a,  platinum;  b,  silver  bromide; 
c,  potassium  sulfate.  Find  the  percentage  deviations  of  these  values 
from  the  measured  values,  which  are,  a,  0.032;  b,  0.074;  c,  0.190. 


Calc. 

Meas. 

Substance                          Calc. 

Meas. 

8.6 

9.1 

PbN2O6                   42.6 

38.8 

24.4 

20.5 

CaSiOs                     22.0 

21.3 

24.4 

25.6 

K4Fe(CN)6              79-o 

79.0 

28.6 

28.7 

CuSO4                     27.6 

25.8 

12.4 

12.4 

CuSO4.5H2O            70.6 

72.7 

18.6 

18.2 

AlK(S04)2.i2H2O  158. 

I65. 

2O.O 

20.  2 

NH4C1                     21.6 

19.9 

24.4 

24.1 

H2C2O4                    24.2 

2S.I 

94         ATOMIC  PROPERTIES  OF  SOLID  SUBSTANCES 

57.  Determination  of  Atomic  Weights.  —  The  law  of  Dulong  ana 
Petit,  even  though  it  is  only  an  approximate  principle,  may  evidently 
be  employed  for  determining  what  multiple  of  the  combining  weight 
of  an  element  is  its  atomic  weight;  and  the  application  of  this  law 
was  in  fact  one  of  the  most  important  methods  by  which  the  present 
system  of  atomic  weight  values  was  established. 

Prob.  2.  — Determination  of  Atomic  Weights.  —  Calculate  the  exact 
atomic  weight  of  an  element  whose  specific  heat-capacity  is  0.092  and 
whose  oxide  contains  88.82%  of  the  element. 


GENERALIZATIONS  RELATING  TO  ATOMIC  WEIGHTS    95 

H.    GENERALIZATIONS   RELATING   TO   ATOMIC  WEIGHTS 

58.  The  Methods  of  Atomic  Weight  Determination  and  the  Periodic 

Law.  —  The  important  methods  already  considered  for  determining 
what  multiple  of  the  combining  weight  is  the  atomic  weight  may  now 
be  briefly  summarized. 

It  was  shown  in  Art.  14  that  the  atomic  weight  of  an  element, 
or  at  any  rate  a  maximum  value  for  its  atomic  weight,  can  be  obtained 
by  finding  the  smallest  weight  of  the  element  contained  in  one  mo- 
lecular weight  of  any  of  its  gaseous  compounds,  as  derived  from  the 
density  and  Avogadro's  principle.  Molecular  weights  derived  from 
the  laws  of  perfect  solutions  (Arts.  36,  38,  49)  may  also  be  used  for 
the  same  purpose. 

It  was  shown  in  Art.  27  that  the  molal  heat-capacity  of  perfect 
gases  is  dependent  on  the  number  of  atoms  in  the  molecule,  that  this 
property  has  a  definite  value  predicted  by  the  kinetic  theory  for  gases 
with  monatomic  molecules,  and  that  therefore  the  atomic  weight  of 
elements  which  form  gases  having  this  molal  heat-capacity  can  be 
immediately  derived  from  their  gas  densities  and  the  corresponding 
molecular  weights. 

It  was  shown  in  Art.  57  that  the  atomic  weight  of  elements  can 
be  directly  determined  from  the  principle  that  the  atomic  heat- 
capacities  of  solid  elementary  substances  at  room  temperature  have 
approximately  the  same  value.  Finally,  atomic  weights  can  also  be 
derived  from  the  phenomenon  of  isomorphism  mentioned  in  Art.  54. 

By  a  combination  of  all  these  methods,  which  have  in  general 
given  concordant  results,  the  present  system  of  atomic  weight  values 
has  been  derived. 

When  the  elements  are  arranged  in  the  order  of  the  so  determined 
atomic  weights,  it  is  found  that  there  is  a  progressive  .change  in  the 
various  properties  of  the  elementary  substances  and  their  compounds, 
and  a  periodic  recurrence  of  similar  properties.  This  generalization 
is  called  the  periodic  law;  and  many  systematic  arrangements  of  the 
elements  have  been  proposed  for  the  purpose  of  bringing  out  as  fully  as 
possible  the  relations  between  their  properties.  These  relations  are 
most  appropriately  considered  in  the  detailed  study  of  inorganic 
chemistry,  and  are  therefore  only  briefly  referred  to  here. 


CHAPTER  V 

THE   ELECTROLYTIC  BEHAVIOR    OF    SOLUTIONS    AND 
THE  IONIC  THEORY 


I.    THE  MOLAL  PROPERTIES   OF   SOLUTIONS   OF   IONIZED   SUBSTANCES 

59.  Effects  of  Salts  on  the  Molal  Properties  of  Aqueous  Solutions 
and  Their  Explanation  by  the  Ionic  Theory.  —  In  dilute  solution  the 
effect  of  salts  of  the  uniunivalent  type,  such  as  sodium  chloride  or 
silver  nitrate,  on  the  vapor-pressure  of  water  and  on  the  other  related 
molal  properties  is  nearly  twice  as  great,  and  the  effect  of  salts  of  the 
unibivalent  type,  such  as  potassium  sulfate  and  barium  chloride,  is 
nearly  three  times  as  great,  as  it  would  be  if  each  formula- weight 
yielded  a  single  mol  in  the  solution.  Strong  acids  and  bases  show  a 
similar  behavior. 

The  following  values  illustrate  the  effect  of  some  typical  ionized 
substances  on  the  freezing-point  of  water  at  a  concentration  o.i 
formal.  The  value  given  is  the  ratio,  called  the  moil-number  (i),  of 
the  effect  produced  by  one  formula- weight  of  the  substance  to  that 
produced  by  one  mol  of  a  perfect  solute. 

NaCl,  1.87;  NaNO3,  1.83;  KC1O3,  1.80;  HC1,  1.92;  MgSO4,  1.32; 

BaCl2,  2.58;  Pb(N03)2,  2.30;  Na2S04,  2.47;  K4Fe(CN)6,  3.32. 
As  the  concentration  decreases  the  value  of  the  mol-number  ap- 
proaches 2  for  the  uniunivalent  substances,  and  3  for  the  unibivalent 
substances. 

These  and  other  facts  have  led  to  the  conclusion  that  these  sub- 
stances are  largely  dissociated  in  aqueous  solution.  For  example, 
NaCl  dissociates  into  Na+  and  Cl~;  HNO3  into  H+  and  NO3~;  H2S04 
into  H+,  H+,  and  SO4=;  and  Ba(OH)2  into  Ba++,  OH~,  and  OH~. 
Tri-ionic  substances  may  also  dissociate  partially  into  intermediate 
ions;  thus  H2SO4  into  HSO4~  (and  H+). 

The  electrical  behavior  of  the  solutions  indicates  that  these  dis- 
sociation-products differ  from  ordinary  substances  in  that  their  mole- 
cules are  electrically  charged.  These  charged  molecules  are  called' 
ions;  and  to  their  formulas  +  or  —  signs  are  attached,  as  in  the  above 
examples,  to  indicate  the  nature  and  magnitude  of  the  charge.  The 
fraction  of  the  salt  dissociated  is  called  its  ionization  y.  This  fraction 
always  decreases  with  increasing  concentration. 

96  * 


MOLAL  PROPERTIES  OF  IONIZED  SUBSTANCES       97 

From  the  effect  of  ionized  substances  on  the  molal  properties  their 
ionization  has  often  been  calculated.  The  evidence,  however,  seems 
to  show  that  ions  produce  on  the  vapor-pressure,  boiling-point,  and 
freezing-point  of  water,  even  at  moderate  concentrations  such  as  o.i 
molal,  effects  which  differ  considerably  from  those  produced  by  perfect 
solutes  at  the  same  molal  concentration.  These  deviations,  which 
will  be  more  fully  considered  in  Art.  79,  make  unreliable  the  exact 
computation  of  the  ionization  of  salts  from  the  vapor-pressure  and 
freezing-point  lowering.  In  spite  of  the  deviations  a  rough  estimate 
of  the  number  of  mols  of  solute  present  in  solutions  of  ionized  sub- 
stances can  be  derived  from  the  results  of  measurements  of  the  molal 
properties,  with  the  aid  of  the  laws  of  perfect  solutions;  and  such 
measurements  often  furnish  useful  information  as  to  the  forms  in 
which  substances  exist  in  the  solution. 

Prob.  i.  —  Effect  of  Salts  on  the  Molal  Properties.  —  A  solution  of  0.65 
formula- weight  of  KC1  in  1000  g.  of  water  has  at  100°  a  vapor-pressure 
of  744.8  mm.  a.  Calculate  the  mol-number  i,  which  expresses  the 
ratio  of  the  observed  lowering  to  that  which  would  be  produced  by  0.65 
mol  of  a  perfect  solute  in  1000  g.  of  water,  b.  Find  the  ionization  of 
the  salt  to  which  this  value  of  i  would  correspond  if  the  ions  acted  as 
perfect  solutes. 

Prob.  2.  —  Determination  of  the  Chemical  Substances  Present  in 
Solutions.  —  The  freezing-point  of  a  solution  containing  0.05  formula- 
weights  of  H2S04  per  1000  g.  of  water  is  -o.  2 1 5°.  a.  Find  the  correspond- 
ing value  of  the  mol-number  i.  b.  Derive  from  this  value  an  estimate 
of  the  secondary  ionization  of  the  HSO4~  into  H+  and  SO4=,  assuming 
that  the  primary  ionization  of  the  H2S04  into  H+  and  HS04~  is  complete 
and  that  the  ions  act  as  perfect  solutes. 


98  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

II.  ELECTROLYSIS  AND  FARADAY'S  LAW 

60.  Electrolytic  Conduction.  —  Conductors  are  divided  into  two 
classes  with  reference  to  the  changes  that  are  produced  in  them  by 
the  passage  of  electric  currents.     Those  which  undergo  no  changes 
except  such  as  are  produced  by  a  rise  in  temperature  are  called  metallic 
conductors.     Those  in  which  the  passage  of  a  current  is  attended  by  a 
chemical  change  are  called  electrolytes.     Aqueous  solutions  of  salts, 
bases,  and  acids,  and  melted  salts  at  high  temperatures,  are  the  most 
important   classes   of   the  well-conducting  electrolytes.     The  most 
obvious  chemical  changes  attending  the  passage  of  a  current  through 
an  electrolyte  are  those  that  take  place  at  the  surfaces  of  the  metallic 
conductors  where  the  current  enters  and  leaves  the  electrolyte.     The 
production  of  such  chemical  changes  by  a  current  from  an  external 
source  is  called  electrolysis.     The  occurrence  of  such  changes,  when 
they  themselves  give  rise  to  an  electric  current,  is  called  voltaic  action. 
Those  portions  of  the  metallic  conductors  that  are  in  contact  with  the 
electrolyte  are  called  the  electrodes;  the  one  at  which  the  current 
leaves  the  electrolyte,  or  the  one  towards  which  the  positive  electricity 
flows  through  the  electrolyte,  is  designated  the  cathode;  the  other, 
the  anode. 

61.  Chemical  Changes  at  the  Electrodes.  —  The  chemical  change 
produced  at  the  cathode  is  always  a  reduction;  that  at  the  anode,  an 
oxidation.    The  following  table  exemplifies  types  of  chemical  chajiges- 
commonly  occurring  in  electrolysis.     It  shows  the  products  resulting 
when  aqueous  solutions  of  certain  typical  salts,  bases,  and  acids  are 
electrolyzed  between  electrodes  which  are  unattacked,  for  example, 
between  platinum  or  carbon  electrodes. 


THE  PRODUCTS  OF  ELECTROLYSIS 

Solute 

Cathode  products 

Anode  products 

Cu(N03)2  or  AgN03 

Cu  or  Ag 

O2  and  HN03 

KNO, 

H2  and  KOH 

O2  and  HNO3 

Na2SO4 

H2  and  NaOH 

O2  and  H2SO4 

KOH  or  Ba(OH)2 

H2 

O2 

H2SO4  or  H3PO4 

H2 

O2 

Dilute  HC1  or  HNO3 

H2 

I  Oj 

Concentrated  HC1 

H2 

C12 

When  the  anode  is  a  metal  which  can  react  with  the  anion  of  the 
solute,  the  change  at  the  anode  may  consist  only  in  the  dissolving  of 
the  metal;  thus  when  a  nitrate  or  sulfate  is  electrolyzed  with  a 


ELECTROLYSIS  AND  FARADAY'S  LAW  99 

copper  anode,  copper  passes  into  solution,  forming  copper  nitrate  or 
sulfate. 

When  the  cathode  is  coated  with  a  reducible  solid  substance,  such  as 
silver  chloride  or  lead  dioxide,  or  is  surrounded  with  a  solution  con- 
taining a  reducible  solute,  such  as  a  ferric  salt  or  a  chromate,  this 
substance  is  reduced  and  there  may  be  no  hydrogen  evolved.  In  any 
definite  electrolytic  cell,  however,  the  nature  of  the  changes  at  the 
electrodes  often  varies  with  the  conditions  of  the  electrolysis,  such  as 
the  applied  electromotive  force,  current-density,  concentration,  and 
temperature. 

In  the  case  of  voltaic  actions,  the  chemical  changes  are  of  a  similar 
character,  most  commonly  consisting  in  the  solution  of  the  metal 
composing  the  anode,  and  in  the  deposition  of  another  metal  or  of 
hydrogen  on  the  cathode  or  in  the  production  of  a  soluble  reduced 
substance  at  the  cathode.  Thus,  in  the  J^anieJL  cell,  which  consists 
of  a  copper  electrode  in  a  copper  sulfate  solution  and  of  a  zinc  electrode 
in  a  zinc  sulfate  solution,  the  two  solutions  being  in  contact  and  the 
two  electrodes  connected  by  a  metallic  conductor,  the  zinc  dissolves 
and  the  copper  precipitates;  and,  in  the  Bunsen  cell,  consisting  of  a 
zinc  electrode  in  dilute  sulfuric  acid  and  a  carbon  electrode  in  strong 
nitric  acid,  zinc  dissolves  at  the  anode,  and  the  hydrogen  primarily 
produced  at  the  cathode  reduces  the  nitric  acid  to  lower  oxides  of 
nitrogen. 

Prob.  3.  —  Changes  at  the  Electrodes.  —  State  what  substances  are 
produced  or  destroyed  at  each  electrode  when  electricity  passes  through: 
o,  a  concentrated  solution  of  NaCl  between  a  carbon  anode  and  an  iron 
cathode;  b,  a  solution  of  NaCl  between  a  silver  anode  and  a  metal 
cathode  coated  with  AgCl;  c,  dilute  H2SO4  between  a  lead  anode  and  a 
cathode  of  lead  coated  with  PbO2;  d,  a  solution  of  ZnSO4  between  a 
zinc-amalgam  anode  and  a  cathode  of  mercury  covered  with  Hg2SO4; 
e,  through  a  solution  of  K2SO4  between  platinum  electrodes. 

62.  Faraday's  Law.  —  The  passage  of  electricity  through  an  elec- 
trolyte is  attended  at  each  electrode  by  a  chemical  change  involving 
a  number  of  chemical  equivalents  N  strictly  proportional  to  the  quan- 
tity of  electricity  Q  passed  through  and  dependent  on  that  alone. 
That  is:  Q  =  FN,  where  r  is  a  constant  with  respect  to  all  variations 
of  the  conditions,  such  as  temperature,  concentration,  current-strength, 
current-density,  etc.  Such  variations  often  influence  the  character 
of  the  chemical  change,  but  not  the  total  number  of  equivalents 


ioo  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

involved.  The  law  is  applicable  to  concentrated,  as  well  as  to  dilute 
solutions,  and  to  fused  salts.  It  is  known  as  Faraday's  law. 

The  constant  F  evidently  represents  the  quantity  of  electricity  pro- 
ducing a  chemical  change  involving  one  equivalent.  It  is  called  a 
faraday,  and  has  the  value  96,500  coulombs.  One  coulomb  is  the  quan- 
tity of  electricity  flowing  per  second  when  the  current  is  one  ampere. 

The  term  chemical  equivalent  in  the  above  statement  of  Faraday's 
law  signifies  the  oxidation  or  reduction  equivalent  of  the  substance 
in  the  sense  in  which  it  is  used  in  volumetric  analysis.  That  is,  one 
equivalent  of  any  substance  is  that  weight  of  it  which  is  capable  of 
oxidizing  one  atomic  weight  of  hydrogen,  or  which  has  the  same 
reducing  power  as  one  atomic  weight  of  hydrogen. 

Applications  of  Faraday's  Law.  —  -  \ 

</  Prob.  4.  —  Through  solutions  of  AgNO3,  BiCl3,  Hg2SO4,  and  PbCl2, 
placed  in  series,  9650  coulombs  are  passed.  How  many  grams  of  metal 
will  be  deposited  on  the  cathode  from  each  solution? 

Prob.  -5.  —  A  Daniell  cell,  consisting  of  zinc  in  ZnS04  solution  and 
copper  in  CuSCX  solution,  furnishes  a  current  of  o.i  ampere  for  ioo 
minutes.  How  many  grams  of  copper  deposit  and  of  zinc  dissolve  in 
the  cell? 

^  Prob.  6.  —  How  long  must  a  current  of  5  amperes  be  passed  through 
dilute  sulphuric  acid  in  order  to  produce  at  27°  and  i  atm.,  a,  one  liter 
of  oxygen?  b,  one  liter  of  hydrogen?  Ans.  a,  52.3  min. 

Prob.  7.  —  1930  coulombs  are  passed  through  a  solution  of  CuSO4. 
At  the  cathode  0.018  equivalent  of  copper  is  deposited.  How  many 
equivalents  of  hydrogen  are  set  free? 

63.  The  Mechanism  of  Conduction  between  Electrodes  and 
Solutions.  —  Faraday's  law  evidently  shows  that  electricity  is  trans- 
ferred from  solution  to  electrode,  or  in  the  reverse  direction,  only  by 
the  constituents  of  the  electrolyte;  and  that  one  equivalent  of  any 
constituent  carries  the  same  quantity  of  electricity,  namely,  one 
faraday  or  96500  coulombs.  These  current-carrying  constituents, 
called  the  ion-constituents,  exist,  according  to  the  ionic  theory,  partly 
combined  with  one  another  as  substances  whose  molecules  are  electri- 
cally neutral,  and  partly  as  separate  substances  whose  molecules  are 
electrically  charged.  These  charged  molecules,  and  also  the  substances 
consisting  of  them,  are  called  ions,  —  the  positively  charged  ones 
cations,  and  the  negatively  charged  ones  onions.  When  it  is  necessary 
to  distinguish  between  the  charged  substance  and  the  charged  mole- 
cule, the  terms  ion-substance  and  ion-molecule  will  be  used. 


ELECTROLYSIS  AND  FARADAY'S  I  LAW ',     :  :-. 


Faraday's  law  gives  no  indication  as  to  which  of  the  ions  that  may 
be  present  in  the  solution  are  involved  in  the  chemical  changes  at  the 
electrodes.  It  is  not  in  general  true  either  that  only  one  kind  of  ion 
or  that  all  the  kinds  of  ions  present  are  discharged  or  produced  at  the 
electrodes.  Even  those  ions  which  are  most  concentrated  in  the 
solution  may  not  be  involved  at  all  in  the  chemical  change  at  the 
electrode.  Thus  in  electrolyzing  a  concentrated  sodium  chloride 
solution,  there  are  discharged  at  the  cathode  not  the  sodium  ions  of 
the  salt,  but  the  hydrogen  ions  of  the  water,  even  though  these  have 
only  a  very  slight  concentration  in  the  solution.  Moreover,  the 
transfer  of  electricity  to  the  cathode  may  be  effected,  not  only  by  the 
deposition  of  a  cation,  but  also  by  the  formation  of  an  anion,  as  in  the 
case  of  an  iodine  electrode;  for  flow  of  negative  electricity  in  one 
direction  produces  a  current  having  the  same  electrical  effects  as 
flow  of  positive  electricity  in  the  other.  The  factors  that  determine 
which  ions  are  involved  in  the  electrode  processes  will  be  considered 
in  Chapter  XI. 

Since  one  mol  of  any  ion-substance  has  a  charge  of  96,500  coulombs 
or  some  simple  multiple  of  this  quantity  of  electricity,  and  since  it 
consists  of  a  definite  number  of  molecules,  the  Avogadro  number, 
6.06  Xio23  (Art.  19),  any  ion-molecule  must  have  a  charge  which  is 
equal  to  or  a  simple  multiple  of  a  definite  quantity  of  electricity, 
namely,  96,5oo/ (6.06 X io23)  orj.SQXio"19  coulomb.  This  is  evidently 
a  quantity  of  electricity,  which,  like  an  atom  of  an  element,  is  not 
subdivided  in  chemical  processes.  This  quantity  of  negative  elec- 
tricity is  called  a  negative  electron,  or  often  simply  an  electron;  the 
corresponding  quantity  of  positive  electricity  is  called  a  positive 
electron. 

Negative  electrons  are  capable  of  existing  as  separate  particles; 
thus  cathode  rays,  and  the  beta  rays  emitted  by  radioactive  sub- 
stances, have  been  shown  to  consist  of  a  stream  of  such  electrons. 
But  positive  electrons  are  known  only  in  association  with  atoms  or 
molecules.  Since  only  the  negative  electron  is  known  to  detach 
itself  from  matter,  anions  are  regarded  as  combinations  of  neutral 
atoms  or  atom-groups  with  one  or  more  negative  electrons,  and  cations 
as  neutral  atoms  or  atom-groups  which  have  lost  one  or  more  negative 
electrons.  Moreover,  since  there  is  no  transfer  of  matter  in  metallic 
conduction,  the  current  is  carried  through  the  metal  solely  by  negative 
electrons. 


';;':       ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

In  accordance  with  these  views,  the  mechanism  of  conduction 
between  electrodes  and  solutions  consists  in  a  passage  of  ions  to  or  from 
the  electrode  and  the  taking  up  or  giving  off  by  the  electrode  of  nega- 
tive electrons,  these  being  continuously  removed  or  supplied  at  its 
surface  by  their  flow  through  the  metallic  conductor.  Thus  in  the 
electrojysis  of  hydrochloric  acid,  hydrogen  ions  move  from  the  solution 
to  the  cathode  and  takeup  there  negative  electrons'  thereby  producing 
neutral  hydrogen  molecules;  and  chloride  ions  move  from  the  solution 
to  the  anode  and  give  off  there  negative  electrons,  thereby  producing 
neutral  chlorine  molecules.  Similarly  in  the  Daniell  cell,  neutral 
zinc  atoms  in  the  anode  give  up  to  it  negative  electrons  and  pass  into 
the  solution  in  the  form  of  zinc  ions,  and  copper  ions  move  to  the 
cathode  from  the  solution,  and  take  up  negative  electrons  from  the 
cathode,  thereby  becoming  neutral  copper  atoms. 

The  chemical  changes  and  the  accompanying  electrical  effects 
taking  place  at  the  electrodes  in  the  way  just  described  may  be  repre- 
sented by  electrochemical  equations,  in  which  a  free  negative  electron 
in  the  electrode  is  represented  by  the  symbol  ©.  Thus,  the  electro- 
chemical equations  for  the  processes  considered  above  are: 

2H+    +     2©       =     H2;  2C1-     =     C12       +       20. 

(in  the  (in  the  (on  the  (in  the  (on  the  (in  the 

solution)          electrode)          electrode)  solution)  electrode)  electrode) 

Zn      =  Zn++     +    2©;  Cu++    +     20       =       Cu. 

(in  the  (in  the  (in  the  (in  the  (in  the  (in  the 

electrode)  solution)          electrode)  [solution)          electrode)  electrode) 

Since  it  has  long  been  customary  to  consider  an  electric  current  as 
a  flow  of  positive  electricity,  it  is  usually  more  convenient  to  express 
the  effects  at  the  electrodes  as  if  they  involved  positive  electrons, 
which  are  denoted  by  the  symbol  0.  Thus  the  above  electrode 
effects  are  expressed  by  the  equations: 

2H+=H2+20;  and2Cl-+20=C!2. 
Zn  +  20  =  Zn++;  and  Cu++=  Cu  -f  20. 

Furthermore,  it  is  usually  advantageous  to  write  the  equations  so  as 
to  show  the  substances  that  are  actually  produced  or  destroyed,  and 
iiot  merely  the  ions  that  are  directly  concerned  in  the  mechanism 
of  the  electrode  process.  Thus  the  cathode  process  in  the  electrolytic 
decomposition  of  water  (in  the  presence  of  a  sodium  salt)  would  be 
expressed  by  the  equation  2H2O  =  H2+20H~+20,  and  not  simply  by 
the  equation  2H+  =  H2-|-20.  In  these  electrochemical  equations, 


ELECTROLYSIS  AND   FARADAY'S   LAW  103 

as  in  other  chemical  equations,  the  formulas  may  represent  either  the 
molecules  and  electrons,  or  the  mols  of  the  substances  and  the  faradays 
of  electricity. 

Prob.  8.  —  Formulation  of  Cells  and  Electrochemical  Equations.  — 

a.  Formulate  (as  in  the  answer  given  below)  the  five  cells  described 
in  Prob.  3,  showing  also  the  substances  produced  at  the  electrodes. 

b.  Write  electrochemical  equations  expressing  the  chemical  changes  and 
electrical  effects  at  each  of  the '  electrodes  when  one  faraday  passes 
through  each  of  these  cells.    Ans.  First  cell:  a,  C+C12,  Na+Cl~  (cone.) 
in  H20,  H2+Fe.    b,  Cl-+0  =  JC12; 


IO4 


ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 


I 


C 


III.    ELECTRICAL    TRANSFERENCE 

64.  Phenomenon  of  Electrical  Transference.  —  When  a  current  is 
passed  through  a  solution  of  a  salt,  base,  or  acid,  in  addition  to  the 
chemical  changes  taking  place  at  the  electrodes  in  accordance  with 
Faraday's  law,  a  certain  quantity  of  the  cation-constituent  is  trans- 
ferred from  the  neighborhood  of  the  anode  to  that  of  the  cathode,  and 
a  certain  quantity  of  the  anion-constituent  is  transferred  in  the  reverse 
direction.  This  phenomenon  can  best  be  made  clear  by  the  considera- 
tion of  an  actual  transference  determination.  Consider,  for  example, 
that  a  0.02  normal  solu^iojnjjLsodium  sulfate  is  electrolyzed  at  18°  in 
an  apparatus  like  that  shown  in  the  figure, 
between  a  platinum  cathode  (marked—)  and  a 
platinum  anode  (marked  +).  To  avoid  stir- 
ring of  the  solution,  the  electrodes,  from  which 
hydrogen  and  oxygen  gases  are  evolved,  are 
placed  near  the  surface;  the  anode,  around 
which  the  solution  becomes  denser  during  the 
electrolysis,  is  placed  near  the  bottom  of  the 
tube;  and  the  apparatus  is  immersed  in  a 
water-bath  kept  at  constant  temperature. 
The  current  is  stopped  before  the  hydroxide- 
,ion  produced  at  the  cathode  and  the  hydro- 
gen-ion produced  at  the  anode  have  been 
transferred  beyond  the  dotted  lines  in  the 
figure.  The  three  portions  of  the  solution 
'  (called  the  cathode-portion,  middle-portion, 
and  anode-portion,  and  marked  C,  M,  and  A, 

respectively)  are  then  separately  removed  from  the  apparatus,  and 
submitted  to  analysis.  The  quantity  of  sodium  and  of  sulfate  present 
in  each  portion  is  compared  with  the  quantity  of  it  originally  associated 
with  the  weight  of  water  contained  in  the  portion.  It  is  found,  if 
the  experiment  has  been  successful,  that  the  middle-portion  has 
undergone  no  change  hi  composition,  that  the  cathode-portion  has 
increased  its  sodium-content  and  decreased  its  sulfate-content,  and 
that  the  anode-portion  has  increased  its  sulfate-content  and  decreased 
its  sodium-content.  It  is  found,  per  faraday  of  electricity  passed 
through  the  solution,  that  the  sodium-content  has  increased  in  the 
cathode-portion  by  0.39  equivalent  and  has  decreased  in  the  anode- 
portion  by  the  same  amount,  and  that  the  sulfate-content  has  in- 


ELECTRICAL   TRANSFERENCE  105 

Creased  in  the  anode-portion  by  0.6 1  equivalent  and  has  decreased 
in  the  cathode-portion  by  the  same  amount. 

Prob.  g.  —  The  Resultant  Effects  of  Transference  and  Electrolysis.  — 
a.  Write  electrochemical  equations  expressing  the  changes  produced  by 
the  electrolysis  at  each  electrode  in  the  experiment  with  sodium  sul- 
fate  described  above,  and  tabulate  the  number  of  equivalents  of 
each  ion-constituent  produced  per  faraday  by  the  electrolysis  both 
in  the  cathode -portion  and  the  anode-portion,  b.  Tabulate  also  the 
gain  and  loss  caused  by  the  transference  per  faraday  in  the  number  of 
equivalents  of  each  ion-constituent  in  the  cathode-portion  and  in  the 
anode-portion,  c.  Tabulate  the  resultant  effect  of  the  electrolysis 
and  transference  on  the  gain  and  loss  in  the  number  of  equivalents  of 
sodium  sulfate,  sodium  hydroxide,  and  sulfuric  acid  in  the  cathode- 
portion  and  in  the  anode-portion. 

When  either  of  the  constituents  whose  transference  is  being  deter- 
mined is  deposited  on  or  dissolved  off  the  electrode,  as  is  the  case  when 
a  silver  nitrate  solution  is  electrolyzed  between  silver  electrodes,  the 
quantity  of  it  so  deposited  or  dissolved  must  evidently  be  determined 
either  by  direct  weighing  or  by  calculation  with  the  aid  of  Faraday's 
law,  and  be  subtracted  from  or  added  to  the  change  in  content  of  that 
constituent  in  the  electrode  portion. 

65.  Law  of  Transference.  —  The  sum  of  the  number  of  equivalents 
of  the  cation  and  anion  constituents  (NC  and  NA)  transferred  in  the 
two  directions  is  equal  to  the  number  of  faradays  (N)  passed  through 
the  solution.    That  is,  Nc+  NA  =  N.    This  law,  called  the  law  of  trans- 
ference, supplementing  Faraday's  law  of  electrolysis,  is  illustrated 
by  the  data  for  sodium  sulfate  given  above. 

In  the  case  of  mixtures  containing  various  ion-constituents  (Ci, 
C2,. .  .Ai,  A2. . .)  all  of  these  are  transferred,  and  the  expression  of  the 
law  of  transference  is:  NCl  +  NC2  . .  +  NAl  +  NA2 . .  =  N. 

66.  Transference-Numbers.  —  The  equivalents  Nc  of  cation-con- 
stituent transferred  are  in  general  not  equal  to  the  equivalents  NA  of 
anion-constituent  transferred.     The  experimentally  determined  num- 
ber of  equivalents  of  any  ion-constituent  transferred  ger  faraday  of 
electricity,  whether  in  a  solution  of  a  single  salt  or  a  mixture  of  salts,  is 
called  the  transference-number  T  of  that  constituent.     Thus  in  the  case 
of  sodium  sulfate  TNa  =  0.39  and  TSo«  =  0.61.     In  accordance  with  the 
law  of  transference  the  sum  of  the  transference-numbers  of  aU  the 
ion-constituents  is  equal  to  unity. 


io6  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

Determination  of  Transference-Numbers.  — 

Prob.  10.  —  Through  a  o.i  formal  solution  of  potassium  sulfate 
between  platinum  electrodes  0.0075  faraday  is  passed  at  25°.  The 
cathode-portion  after  the  electrolysis  is  found  to  contain  0.1450  g.  more 
potassium  than  was  originally  associated  with  the  weight  of  water  in 
the  portion.  Calculate  the  transference-number  of  the  sulfate-ion. 

Prob.  ii. — A  current  is  passed  at  25°  through  a  solution  of  i6.64g.  of 
Pb(NO3)2  in  1000  g.  of  water  between  lead  electrodes  until  0.1658  g.  of 
silver  is  deposited  in  a  coulometer  in  series  with  it.  The  anode  portion 
weighed  62.50  g.  and  yielded  on  analysis  i.i 23  g.  of  PbCrO4.  Calculate 
the  transference-number  of  the  lead-ion.  Assume  that  lead  dissolves 
off  the  anode  in  accordance  with  Faraday's  law.  Ans.  0.489. 

67.  The  Mechanism  of  Conduction  in  Solutions.  —  Just  as  Fara- 
day's law  shows  that  electricity  is  carried  from  the  solution  to  the 
electrode  only  by  the  ion-constituents,  so  the  law  of  transference  shows 
that  through  the  solution  the  electricity  is  likewise  carried  only  by  the 
ion-constituents.  The  only  difference  is  that  often  only  one  kind  of 
ion-constituent  carries  the  electricity  to  the  electrode,  while  all  the 
ion-constituents  present  take  part  in  the  conduction  of  it  through 
the  solution. 

That  the  ion-constituents  do  move  through  the  solution  can  be 
shown  by  placing  the  solution  of  a  salt,  such  as  copper  sulfate  or 
potassium  permanganate,  whose  cation  or  anion  has  a  characteristic 
color,  beneath  a  solution  of  a  colorless  salt,  such  as  potassium  sulfate, 
and  applying  a  potential-difference  at  the  electrodes. 

The  movement  of  the  ion-constituents  is  explained  by  the  ionic 
theory  as  follows.  A  certain  fraction  of  the  molecules  of  a  salt  exists 
in  the  state  of  positively  and  negatively  charged  molecules  —  the 
cations  and  anions.  When  a  solution  is  placed  between  electrodes 
that  are  at  different  potentials,  the  ions  in  virtue  of  their  charges 
are  subjected  to  an  electric  force  which  drives  them  through  the  solu- 
tion —  the  cations  towards  the  cathode,  the  anions  towards  the 
anode;  while  the  unionized  molecules,  being  electrically  neutral,  are 
unaffected.  The  ions  are,  however,  constantly  uniting  to  form  union- 
ized molecules,  and  the  latter  are  constantly  dissociating  into  ions. 
For  this  reason  although  at  any  moment  only  the  ions  are  moving 
the  resultant  effect  is  that  the  ion-constituent  as  a  whole  moves  con- 
tinuously towards  the  electrode.  The  rate  at  which  the  ion-constit- 
uent moves  is  equal  to  the  rate  at  which  the  ion  moves  multiplied 
by  the  ionization  of  the  salt;  for  the  statement  that  a  certain  fraction 


ELECTRICAL  TRANSFERENCE  107 

of  the  molecules  is  ionized  is  equivalent  to  the  statement  that  any 
one  molecule  exists  in  the  form  of  its  ions  during  that  fraction  of  the 
time.  The  quantitative  application  of  these  considerations  to  trans- 
ference is  discussed  in  the  following  article.  ^ 

68.  Transference  in  Relation  to  the  Mobility  of  the  Ions.  —  The 
electric  force  /  acting  on  any  charged  body  is  equal  to  its  charge  Q 
multiplied  by  the  potential-gradient;  that  is,  /=Q  (dv /dl),  where 
d&  is  the  change  of  potential  in  the  distance  dl.  .  Moreover,  the  velocity 
of  any  body  moving  through  a  medium  of  great  frictional  resistance 
is  proportional  to  the  force  acting  upon  it.  Therefore,  since  the 
resistance  to  the  motion  of  ions  through  solutions  is  very  great,  the 
velocity  u  of  any  given  ion  is  proportional  to  the  potential-gradient; 
that  is:  u=u(dv/dl),  where  u  is  the  velocity  under  unit  potential- 
gradient,  called  the  mobility  of  the  ion.  The  velocity  of  the  ion- 
constituent  is  evidently  also  proportional  to  the  potential-gradient; 
and  its  velocity  under  unit  potential-gradient  is  called  the  mobility  of 
the  ion-constituent. 

As  shown  in  the  following  problem,  the  ratio  of  the  mobilities  of 
the  two  constituents  is  equal  to  the  ratio  of  the  transference-numbers^ 
and  the  fraction  of  the  current  carried  by  each  constituent  is  equal 
to  its  transference-number. 

Prob.  12.  —  Transference  in  Relation  to  Ion-Mobility.  —  A  o.i  normal 
solution  of  NaCl  at  18°  is  placed  in  a  vertical  cylindrical  tube  2  cm.  in 
diameter  between  electrodes  20  cm.  apart.  At  the  upper  electrode  a 
potential-difference  of  50  volts  is  applied,  and  the  concentration  changes 
at  the  electrodes  are  eliminated  so  that  the  potential-gradient  remains 
uniform  throughout  the  tube.  The  mobilities  of  the  sodium  and 
chloride  ion-constituents  in  this  solution  at  18°  have  been  found  to  be 
0.000373  and  0.000578  cm.  per  second,  respectively,  a.  Sketch  the 
apparatus  approximately  to  scale.  Calculate  the  distance  which  each 
of  the  ion-constituents  moves  in  30  minutes,  and  locate  these  distances 
in  the  sketch  with  reference  to  a  cross-section  indicated  in  the  middle 
part  of  the  tube.  b.  Calculate  the  number  of  equivalents  of  each 
ion-constituent  transferred  through  any  cross-section  in  30  minutes. 

c.  Calculate  the  transference-numbers  of  the  two  ion-constituents,  and 
show  that  the  ratio  of  them  is  equal  to  the  ratio  of  their  mobilities. 

d.  Calculate  the  current  due  to  the  migration  of  each  ion-constituent, 
and  show  that  its  transference-number  is  equal  to  the  fraction  of  the 
current  carried  by  it.    Ans.  b,  0.000527  equivalent  of  Na. 

Prob.  13.  — Derivation  of  the  Mobility  of  Ion-Constituents  from  Trans- 
ference Experiments.  —  An  ordinary  transference  determination  is 
made  at  18°  with  o.i  normal  AgNO8  solution  in  a  vertical  tube  4  cm.  in 


108  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

diameter  between  silver  electrodes  30  ccm.  apart.  Analysis  of  the 
anode-portion,  which  is  withdrawn  at  the  bottom,  shows  that  0.00207 
equivalent  of  silver  has  migrated  out  of  it.  a.  Sketch  the  apparatus, 
indicating  a  boundary  of  the  anode-portion  and  a  boundary  of  a  region 
above  it  which  contains  all  the  silver  which  was  transferred  out  of  the 
anode  portion,  b.  Calculate  the  distance  through  which  the  silver 
migrated  during  the  passage  of  the  current,  referring  to  the  sketch  for 
the  relations  involved,  c.  The  potential-difference  applied  at  the 
electrodes  was  10  vojts,  and  the  concentration  changes  at  the  electrodes 
were  eliminated  so  that  the  potential-gradient  was  uniform.  The 
resulting  current  of  0.0395  ampere  passed  for  3  hours.  Calculate  the 
mobility  in  centimeters  per  second  of  each  of  the  ion-constituents. 
Ans.  c,  0.000520  for  the  NO«. 

69.  The  Moving-Boundary  Method  of  Determining  Transference.  — 
A  direct  method  of  determining  transference-numbers  is  based  on 
the  proportionality  between  ion-mobility  and  transference  considered 
in  Art.  68.  In  this  method,  which  is  illustrated  by  the 
figure,  the  relative  rates  are  measured  at  which  the  two 

f1  A 

boundaries  of  a  solution  of  a  salt  CA  move  when  placed 


CA 


CA' 


between  solutions  of   two  other  salts  C'A  and  CA', 
;    arranged  as  in  the  figure,  in  which  cc  and  aa  represent 
the  original  positions  of  the  boundaries,  c'c'  and  a' a' 
their  positions  after  a  certain  time.    It  is  evident  that 
the  cation-constituent  C  moves  the  distance  cc'  while  the 
anion-constituent  A  moves  the  distance  aa' \  and  that 
they  are  moving  under  the  same  potential-gradient,  since 
they  are  in   the  same  solution.     Therefore  the  ratio 
ccf/aaf  is  the  ratio  of  the  mobilities  UC/UA,  and  hence  of 
the  transference-numbers  TC/TA. 
The  boundaries  are  most  readily  seen  when  the  "indicator"  ions 
C'  and  A'  are  colored;  but  even  when  the  ions  are  all  colorless,  the 
boundaries  are  usually  visible  because  of  the  different  refractive 
power  of  the  adjoining  solutions. 

Prob.  14.  —  Experimental  Determination  of  Transference-Numbers 
by  the  Moving-Boundary  Method.  —  In  a  moving-boundary  experiment 
an  apparatus  like  that  represented  in  the  figure  is  charged  at  18°  with 
solutions  of  silver  nitrate  at  the  bottom,  of  potassium  nitrate  in  the 
middle,  and  of  potassium  acetate  at  the  top.  The  lower  electrode, 
which  is  of  silver,  is  made  the  anode.  In  90  minutes  the  lower  boundary 
moves  3.00  cm.  and  the  upper  boundary  2.88  cm.  Sketch  the  apparatus, 
indicating  the  positions  of  the  three  solutions  and  the  boundaries 
between  them  at  the  beginning  and  end  of  the  experiment.  State  what 
transference-numbers  can  be  derived,  and  calculate  their  values. 


ELECTRICAL   TRANSFERENCE  109 

*In  order  that  the  boundary  between  the  two  solutions  C'A  and  CA 
remain  sharp,  it  is  essential  that  the  ion  C'  have  a  smaller  mobility 
than  the  ion  C  which  precedes  it.  For  evidently,  if  any  quantity  of 
this  ion  C',  through  migration  or  convection,  should  enter  the  solution 
CA,  it  would  in  consequence  of  its  lesser  mobility  move  in  that  solution 
with  a  smaller  velocity  than  the  ion  C  and  drop  back  to  the  boundary. 
That  the  ion  C'  does  not,  as  a  result  of  its  smaller  mobility,  lag  behind 
and  thus  destroy  the  sharpness  of  the  boundary,  is  due  to  the  facts 
that,  when  it  begins  to  do  so,  its  concentration  in  the  region  behind  the 
boundary  decreases,  and  consequently  the  resistance  of  the  solution 
and  therefore  the  potential-gradient  in  that  region  increases,  which  in 
turn  increases  the  actual  velocity  of  the  ion  and  brings  it  back  to  the 
boundary.  It  can  be  shown  (as  in  Prob.  38,  Art.  80)  with  the  aid  of  the 
principles  correlating  resistance  with  concentration  and  ion-mobilities 
(considered  in  Art.  73)  that  the  concentration  of  the  substance  C'A 
automatically  adjusts  itself  so  that  the  ratio  of  this  concentration  to 
that  of  the  substance  CA  is  equal  to  the  ratio  of  the  transference 
number  of  the  ion-constituent  C'  in  the  solution  C'A  to  that  of  the 
ion-constituent  C  in  the  solution  CA. 

70.   Change  of  Transference-Numbers  with  the  Concentration.  — 

The  following  table  shows  the  transference-numbers  of  some  typical 
substances  at  18°  at  a  series  of  concentrations. 

TRANSFERENCE-NUMBERS  AT  VARIOUS  CONCENTRATIONS 

Equivalent 
per  liter 

O.OO 
0.01 
0.05 
0.10 
0.20 
1. 00 

It  will  be  noticed  that  the  transference-number  in  the  case  of  po- 
tassium chloride  does  not  change  appreciably  with  decreasing  concen- 
tration below  o.i  normal.  It  is  therefore  to  be  inferred  that  the 
transference-number  will  have  this  same  value  (0.504)  as  the  con- 
centration approaches  zero,  as  shown  in  the  table.  The  value  at  zero 
concentration  evidently  corresponds  in  general  to  a  definite  ratio  of 
the  mobilities  of  the  ions  in  pure  water;  and  an  accurate  knowledge 


Anion  Transference-Numbers 

KCl 

HCl                         NaCl 

0.504 

0.172                    0.602 

0.504 

0.167                    0.604 

0.504 

0.166              0.605 

0.505 

0.165                    0.607 

0.506 

0.163               0.610 

— 

0.156                    0.631 

no  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

of  one  such  value  is  of  importance,  since  it  enables  the  conductances 
of  all  the  various  ions  to  be  evaluated,  as  will  be  shown  in  Art.  76. 
The  transference-numbers  given  in  the  table  for  sodium  chloride  and 
hydrochloric  acid  at  zero  concentration  are  not  the  results  of  trans- 
ference measurements,  but  were  derived  conversely  from  the  con- 
ductances of  the  separate  ions. 

The  transference-numbers  for  sodium  chloride  and  hydrochloric 
acid  change  considerably  with  the  concentration  even  below  0.2 
normal.  Thus  the  change  in  the  transference-numbers  in  passing 
from  0.2  normal  to  o.o  normal  corresponds  to  a  change  in  the  ratio 
of  the  mobilities  of  the  chloride  and  sodium  ions  from  1.56  to  1.51 
(or  3.2%)  and  in  the  ratio  of  the  mobilities  of  the  hydrogen  and  chloride 
ions  from  5.14  to  4.81  (or  6.4%).  This  change  in  the  relative  mobilities 
of  the  two  ions  has  an  important  bearing,  considered  in  Art.  74,  on  the 
change  of  conductance  with  the  concentration;  for  it  shows  that  a 
change  in  the  actual  mobility  of  some  of  the  ions  takes  place  even  at 
these  low  concentrations.  Thus,  if  the  chloride  ion  has  the  same 
mobility  at  the  concentrations  0.2  and  o.oi  normal,  the  sodium  ion 
must  have  a  3.2%  smaller  mobility  in  the  0.2  normal  solution;  but  the 
chloride  ion  also  may  well  have  a  smaller  mobility  at  0.2  than  at  o.oi 
normal,  in  which  case  the  mobility  of  the  sodium  ion  would  decrease 
by  even  more  than  3.2%. 

At  concentrations  above  0.2  normal  the  transference-number  often 
changes  rapidly  with  increasing  concentration.  This  may  be  due  to  a 
variety  of  influences  not  affecting  the  values  at  smaller  concentrations. 
Thus  it  may  arise  from  a  change  in  the  frictional  resistance  of  the 
medium;  from  hydration  of  the  ions,  which  causes  water  to  be  trans- 
ferred and  thus  affects  the  transference  value,  since  this  is  computed 
under  the  assumption  that  the  water  is  stationary;  from  existence  of 
intermediate  ions  or  complex  ions,  which  in  concentrated  solutions 
are  more  likely  to  be  present  in  considerable  quantity.  ^ 

71.  The  Composition  of  Ions  Determined  by  Transference  Experi- 
ments. —  Transference  experiments  furnish  in  many  cases  useful 
information  as  to  the  nature  of  the  ions  present  in  the  solution.  Thus 
they  may  show,  as  illustrated  by  the  following  problems,  the  formation 
of  complex  cations  or  anions  and  the  existence  of  hydrated  ions.  They 
may  also  indicate  the  extent  to  which  salts  are  hydrolyzed  in 
solution. 


ELECTRICAL   TRANSFERENCE  in 

Formation  of  Complex  Ions.  — 

Prob.  15.  —  When  a  current  is  passed  through  a  solution  of  potassium 
silver  cyanide  (KCN.AgCN)  silver  deposits  on  the  cathode  and  the 
cathode-portion  loses  1.40  equivalents  of  silver  and  0.80  equivalent  of 
cyanogen,  and  gains  0.60  equivalent  of  potassium  per  faraday.  a.  Ex- 
plain what  this  shows  in  regard  to  the  composition  of  the  ions  and 
their  transference-numbers,  b.  Write  an  electrochemical  equation 
showing  the  substances  actually  produced  and  destroyed. 

Prob.  16.  —  A  transference  experiment  is  made  by  passing  o.oi  fara- 
day through  a  solution  0.2  formal  in  AgN03  and  0.6  formal  in  NH3 
between  silver  electrodes  at  18°.  The  anode-portion  is  found  to  gain 
0.0053  equivalent  of  silver  and  to  lose  0.0094  formula-weight  of  NH3. 
Explain  what  this  shows  in  regard  to  the  composition  of  the  ions. 
(Silver  dissolves  at  the  anode  in  accordance  with  Faraday's  law.) 

*Prob.  17.  —  The  Hydration  of  Ions.  —  In  a  transference  experiment 
0.0629  faraday  was  passed  at  25°  through  a  solution  placed  between 
a  silver  anode  and  a  silver  cathode  coated  with  silver,  chloride  and 
containing  1.21  formula-weights  of  NaCl  and  o.i  formula- weight  of 
raffinose  (dgH^Oie)  in  1000  g.  of  water.  The  anode-portion  was 
found  by  analysis  to  contain  0.96  g.  less  water  and  1.41  g.  less  NaCl 
than  was  originally  associated  with  the  raffinose  present  in  that  portion. 
a.  Calculate  the  number  of  mols  of  water  and  the  number  of  equivalents 
of  sodium  transferred  per  faraday  from  the  anode  to  the  cathode,  assum- 
ing that  the  raffinose  does  not  migrate,  b.  Assuming  no  hydration 
of  the  chloride  ion,  calculate  the  number  of  molecules  of  water  associated 
with  the  atom  of  sodium  in  the  sodium  ion.  c.  Assuming  the  chloride 
ion  to  be  hydrated  with  x  molecules  of  water,  derive  an  expression  for 
the  hydration  of  the  sodium  ion.  Ans.  c,  2.0  +  1.61  x. 

Transference  experiments,  like  that  described  in  the  preceding 
problem,  have  been  made  with  various  other  chlorides  for  the  purpose 
of  determining  the  hydration  of  the  ions.  These  experiments  have  led 
to  the  following  mean  values  for  the  number  of  molecules  of  water 
contained  in  other  ions,  assuming  the  number  in  the  chloride  ion  to 
be  x  molecules: 

H'+:  0.28  +  0.185  x.  Na+ :  2.0  +  1.61  x. 

Cs+ 10.67  +  1.03*.  Li  +:  4.7  +  2.29*. 

K+:  1.3  +1.023. 


112  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

IV.    ELECTRICAL  CONDUCTANCE 

72.  Conductance,  Specific  Conductance,  and  Equivalent  Conduct- 
ance. —  According  to  Ohm's  law,  the  current  I  flowing  between  two 
planes  of  a  conductor  is  proportional  to  the  potential-difference  E  at 
those  planes.  The  ratio  of  the  current  to  the  potential-difference  is 
called  the  conductance  L;  and  the  inverse  ratio,  the  resistance  R. 
That  is, 

I/E = L,   and   E/I  = R. 

When  the  current  is  expressed  in  amperes  and  the  potential-differ- 
ence in  volts,  the  resistance  is  in  ohms  and  the  conductance  in  recipro- 
cal ohms.  An  ohm  is  the  resistance,  and  a  reciprocal  ohm  (or  mho)  the 
conductance,  of  a  column  of  mercury  at  o°  one  square  millimeter 
hi  cross-section  and  106.3  centimeters  long. 

In  accordance  with  these  definitions,  the  conductance,  expressed 
in  reciprocal  ohms,  of  any  conductor  is  the  number  of  coulombs  which 
pass  per  second  when  the  potential-difference  at  the  ends  is  one  volt. 

The  conductance  of  a  homogeneous  body  of  uniform  cross-section 
is  proportional  to  its  cross-section  a  and  inversely  proportional  to  its 
length  1.  That  is,  L  =  L  a/I.  The  proportionality-factor  L,  which  is 
the  conductance  when  the  cross-section  is  one  square  centimeter  and 
the  length  one  centimeter,  is  called  the  specific  conductance.  Its 
reciprocal  is  called  the  specific  resistance. 

It  has  been  shown  by  transference  measurements  and  by  other  con- 
siderations that  in  aqueous  solutions  of  salts,  acids,  and  bases,  the  ions 
of  the  water  are  present  at  a  concentration  which  is  negligible  even 
when  the  concentration  of  the  ions  of  the  solute  is  as  small  as  o.oooi 
normal.  The  conductance  of  such  solutions  therefore  arises  only  from 
the  motion  of  the  ions  of  the  salt,  acid,  or  base;  and  it  is  convenient  to 
Nemploy  a  concept  by  which  the  conductance  may  be  referred  to  one 
equivalent  of  the  solute.  Such  a  concept  is  equivalent  conductance  A, 
which  is  defined  to  be  the  conductance  of  that  volume  of  solution 
which  contains  one  equivalent  of  salt,  acid,  or  base,  when  placed 
between  parallel  electrodes  one  centimeter  apart.  Since  the  equivalent 
conductance  varies  with  the  volume  of  the  solution  in  which  the 
equivalent  of  salt  is  dissolved,  it  is  necessary  to  specify  the  normal 
concentration  to  which  any  given  value  refers.  Thus,  in  accordance 
with  these  statements,  the  equivalent  conductance  A0.i  of  a  salt  in 
o.i  normal  solution  denotes  the  conductance  of  10,000  cubic  centi- 


ELECTRICAL  CONDUCTANCE  113 

meters  of  that  solution  when  placed  between  parallel  electrodes  one 
centimeter  apart. 

Relations  between  Conductance,  Specific  Conductance,  and  Equivalent 
Conductance.  — 

Prob.  18.  —  Find  the  specific  conductance  in  reciprocal  ohms  of 
mercury  at  o°. 

Prob.  ig.  —  a.  Derive  an  algebraic  relation  expressing  equivalent  con- 
ductance in  terms  of  specific  conductance  and  of  concentration  in 
equivalents  per  cubic  centimeter,  b.  Calculate  the  specific  conduc- 
tances of  o.i  and  o.oi  normal  NaCl  solutions  whose  equivalent  con- 
ductances at  18°  are  92.0  and  101.9  reciprocal  ohms,  respectively. 

Prob.  20.  —  A  o.i  normal  AgNO3  solution  at  18°  is  placed  in  a  tube 
4  cm.  in  diameter  between  silver  electrodes  12  cm.  apart.  A  potential 
difference  of  20  volts  at  the  electrodes  produces  a  current  of  0.1976 
ampere.  Calculate  the  conductance,  the  specific  conductance,  and  the 
equivalent  conductance  of  the  solution. 

Prob.  21.  — The  equivalent  conductance  of  a  o.oi  normal  CuSO4  solu- 
tion at  1 8°  is  7 1 . 7  reciprocal  ohms.  Calculate  the  resistance  of  a  column 
of  it  20  cm.  long  and  5  sqcm.  in  cross-section. 

Prob.  22. — Experimental  Determination  of  Specific  Conductance. — 
In  a  cylindrical  vessel  with  fixed  platinum  electrodes  are  measured 
in  succession  the  resistances  at  18°  of  a  o.oi  normal  KC1  solution,  a 
o.ooi  normal  K2S04  solution,  and  the  distilled  water  used  for  prepar- 
ing these  solutions  (which  has  an  appreciable  conductance  due  to 
impurities,  such  as  carbonic  acid  and  ammonium  hydroxide).  These 
resistances  are  found  to  be  97.8,  937,  and  100,000  ohms,  respectively. 
a.  Find  the  conductances  in  the  vessel  of  the  two  solutions  and  of  the 
water;  also  the  conductances  which  the  two  solutions  would  have 
shown  if  the  water  had  been  free  from  the  impurities  (which  do  not 
react  chemically  with  the  neutral  salts),  b.  From  these  results  and 
the  specific  conductance  at  18°  of  pure  o.oi  normal  KC1  solution, 
which  is  known  to  be  0.001225  reciprocal  ohm,  calculate  the  specific 
conductance  of  pure  o.ooi  normal  K2S04  solution.  Ans.  b,  0.0001268. 

73.  Conductance  in  Relation  to  the  Mobility  of  the  Ion-Constit- 
uents. —  The  discussion  of  the  mechanism  of  conduction  hi  solutions 
in  Arts.  67  and  68,  considered  in  connection  with  the  definition  of 
equivalent  conductance  given  in  Art.  72,  leads  to  the  following  princi- 
ples, which  are  further  illustrated  by  the  problems  below. 

(i)  The^equivalent  condnctance-of  a  substance  at  amz:,concentra- 
tionisJhje_suinjQitiLe.equivaIejit .conductances. of. theionhgonstitujents 
at  that  concenlratiori;  andjthe  transf erence-number  oi  an.  innrcon- 
stituentjs  the  ratio  of  its  equivalent  conductance  to  the  sum  of  the 
equivalent  conductances  of  the  two  ion-constituents.  _  That  is, 


114  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

where  AC  and  AA  are  the 

equivalent  cono*trcta"nces  of  the  cation  and  anion  constituents  at  any 
definite  concentration. 

(2)  The  equivalent  conductance  of  an  ion-constituent  in  any  solu- 
tion is  equal  to  the  product  of  its  mobility  u  in  that  solution  by  the 
^v       cha£gftj^gn  one  equivalent;  that  is,  Ac  =uc  F,  and  AA =UA  F. 

Prob.  23.  —  Equivalent  Conductance  in  Relation  to  the  Mobilities  of 
the  Ion-Constituents.  —  Derive  principle  (2)  stated  above  by  considering 
the  number  of  equivalents  of  each  constituent  that  reaches  the  electrode 
per  second  hi  an  apparatus  corresponding  to  the  definition  of  equivalent 
conductance. 

Prob.  24.  —  Specific  Conductance  in  Relation  to  the  Concentrations 
and  Mobilities  of  the  Ion-Constituents.  —  Derive  an  expression  for  the 
specific  conductance  I  of  a  solution  containing  KC1  at  concentration  Ci 
and  NaNO3  at  concentration  c2  by  considering  the  number  of  equivalents 
of  each  constituent  that  reach  the  electrodes  per  second  in  an  apparatus 
corresponding  to  the  definition  of  specific  conductance. 

74.  Change  of  Equivalent  Conductance  with  the  Concentration.  — 
The  effect  of  the  concentration  on  the  equivalent  conductance  of 
various  types  of  substances  is  illustrated  by  the  following  table. 

EQUIVALENT  CONDUCTANCE  AT  18° 

HCtHafo 

i-34 
4.68 


Equivalents 

per  liter 

NaCl 

K2S04 

HCl 

1.0 

74-3 

71.6 

— 

O.I 

92.0 

94.9 

351-4 

O.OI 

101.9 

115.8 

369-3 

O.OOI 

106.4 

126.9 

375-9 

0.0001 

108.0 

130.7 

378:i 

0.0 

108.5 

132.0 

378.3 

348. 

The  equivalent  conductance  is  seen  in  all  cases  to  increase  with 
decreasing  concentration,  and  in  the  case  of  the  first  three  (largely 
ionized)  substances  to  approach  at  the  lower  concentrations  a  constant 
limiting  value.  This  limiting  value,  which  by  definition  is  the  value 
approached  by  the  conductance  of  one  equivalent  of  substance  between 
electrodes  one  centimeter  apart  as  the  volume  in  which  it  is  dissolved 
approaches  infinity,  is  called  the  equivalent  conductance  at  zero  con- 
centration, and  is  denoted  by  AQ.  This  value  corresponds  to  complete 
ionization,  as  will  be  shown  later. 

In  the  case  of  largely  ionized  substances,  the  AO  value  is  obtained 
by  extrapolation  from  the  values  measured  at  concentrations  between 


ELECTRICAL  CONDUCTANCE  115 

o.i  and  o.oooi  normal.  In  the  case  of  substances  (such  as  acetic  acid) 
that  are  not  largely  ionized,  the  equivalent  conductance  is  far  removed 
from  the  limiting  value  corresponding  to  complete  ionization,  even 
at  the  lowest  concentration  at  which  accurate  measurements  can  be 
made.  For  such  substances  the  A0  value  cannot  be  directly  derived, 
but  is  obtained  with  the  aid  of  the  principles  that  the  equivalent 
conductance  of  a  substance  is  the  sum  of  the  equivalent  conductances 
of  its  ion-constituents,  and  that  at  zero  concentration  any  definite' 
ion-constituent  has  the  same  conductance  whatever  be  the  other  ion- 
constituent  with  which  it  is  associated.  This  last  principle  is  a 
consequence  of  the  facts  that  at  zero  concentration  the  ion-constitu- 
ents are  present  wholly  in  the  form  of  the  (free)  ions,  and  move  through 
the  same  medium  (pure  water),  whatever  be  the  substance  from  which 
they  originate. 

Prob.  25.  —  Derivation  of  Ao  Values  for  Slightly  Ionized  Substances.  — 
a.  Derive  the  Ao  value  at  18°  for  HC2H302  from  the  above  stated 
principles  and  the  following  data.  The  Ao  values  for  HC1,  NaCl,  and 
NaC2H3O2  derived  by  direct  extrapolation  from  observed  data  are 
378.3,  108.5,  and  78.2  respectively,  b.  State  what  Ao  values  obtainable 
by  extrapolation  might  be  used  to  derive  the  Ao  value  for  NH4OH,  a 
slightly  ionized  substance. 

The  conductance  of  slightly  ionized  substances  (such  as  acetic  acid) 
changes  with  the  concentration  in  the  way  that  the  mass-action  law 
requires,  assuming  that  the  conductance  is  proportional  to  the  ioniza- 
tion, as  will  be  considered  in  Art.  75.  But  this  is  not  even  approxi- 
mately true  of  the  conductance  of  largely  ionized  substances  (like 
sodium  chloride,  hydrochloric  acid,  and  potassium  sulfate).  In  the 
case  of  these  substances,  the  fractional  decrease  of  the  equivalent 
conductance  is  closely  proportional  to  the  cube-root  of  the  concentra- 
tion for  uniunivalent  salts  between  o.ooi  and  0.2  normal,  and  some- 
what less  accurately  so  for  unibivalent  salts  between  o.ooi  and  o.i 
normal.  This  empirical  principle  is  expressed  by  the  equations: 


or       = 


In  these  equations  B  is  a  constant  for  any  definite  substance  at  a 
definite  temperature.  Its  value  is  not  greatly  different  for  different 
salts  of  the  same  valence  type,  and  it  varies  only  slightly  with  the 
temperature.  Thus  at  18°  or  25°  its  value  varies  only  between  0.30 
and  0.45  for  uniunivalent  salts,  and  between  0.50  and  0.75  for  uni- 
bivalent salts.  For  largely  ionized  uniunivalent  acids  and  bases  the 


. 


n6  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

value  is  somewhat  smaller  than  for  the  neutral  salts;  thus  for  hydro- 
chloric acid  it  is  0.17.  This  equation  is  approximately  true  for  any 
definite  substance  also  when  other  largely  ionized  substances  are 
present  with  it  in  the  solution,  the  concentration  c  then  denoting  the 
sum  of  the  normal  concentrations  of  all  the  substances. 

*Prob.  26.  —  a.  Plot  the  values  of  the  equivalent  conductance  of 
sodium  chloride  and  of  potassium  sulfate  given  in  the  above  table  as 
ordinates  against  the  cube-root  of  the  concentration  as  abscissas,  in- 
cluding also  the  A  values  at  0.2  normal,  which  are  87.7  for  NaCl  and 
87.7  for  K2SO4;  and  draw  a  smooth  curve  through  the  points,  b.  By 
extending  the  straight-line  portion  of  the  graph  find  the  values  of  Ao 
and  of  the  constant  B  to  which  the  cube-root  equation  leads.  (The 
value  of  Ao  occurring  in  the  cube-root  equation  is  slightly  larger  (about 
i%  for  uniunivalent  salts)  than  that  obtained  by  extrapolating  with 
the  aid  of  functions  which  correspond  more  closely  to  the  observed 
conductance  values  at  concentrations  between  o.ooi  and  o.oooi  normal. 
The  cube-root  equation  seems  to  become  inexact  at  these  very  small 
concentrations). 

The  specific  conductance  of  mixtures  of  largely  ionized  substances 
has  been  found  to  be  determined  by  the  principle  -  that  each  of  the 
substances  has  approximately  that  equivalent  conductance  which  it 
would  have  if  it  were  present  alone  hi  the  solution  at  a  normal  con- 
centration equal  to  the  sum  of  the  normal  concentrations  of  all  the 
substances  present.  With  the  aid  of  this  principle  the  specific  con- 
ductance of  such  mixtures  can  be  calculated;  it  being  noted  that  each 
substance  contributes  to  the  specific  conductance  an  amount  equal  to 
the  product  of  its  equivalent  conductance  in  the  mixture  by  its  normal 
concentration. 

Specific  Conductance  of  Mixtures  of  Largely  Ionized  Substances.  — 
Prob.  27.  —  Find  the  specific  conductance  at  18°  of  a  solution  0.08 
normal  in  NaCl  and  0.02  normal  in  HC1,  referring  to  the  above  table  of 
conductances  for  the  data  needed. 

*Prob.  28.  —  Find  the  specific  conductance  at  18°  of  a  solution  o.io 
normal  in  K2SO4  and  0.05  normal  in  NaCl,  with  the  aid  of  the  plot 
made  in  Prob.  26. 

76.  Conductance  in  Relation  to  the  Concentration  and  Mobility  of 
the  Ions.  —  The  relation  of  conductance  to  the  mobility  of  the  ion- 
constituents  was  considered  in  Art.  73.  It  remains  to  consider  its 
relation  to  the  concentration  and  mobilities  of  the  ions;  for  it  is  the 
motion  of  the  ions  themselves  which  really  constitutes  the  mechanism 
of  conduction. 


ELECTRICAL  CONDUCTANCE  117 

It  has  already  been  explained  in  Art.  67  that  the  mobility  uc  or 
UA  of  an  ion-constituent  at  any  definite  concentration  c  is  equal  to  the 
mobility  uc+  or  UA-  of  the  (free)  ion  at  that  concentration  multiplied 
by  the  ionization  7  of  the  substance;  that  is,  uc  =  7Uc+  and  UA  =  TUA-. 
Combining  this  principle  with  the  principles  (Art.  73)  that  the  equiva- 
lent conductance  of  an  ion-constituent  at  any  definite  concentration  is 
the  product  of  its  mobility  by  the  charge  F  on  one  equivalent,  and  that 
the  equivalent  conductance  of  a  substance  at  any  definite  concentra- 
tion is  the  sum  of  the  equivalent  conductances  of  its  ion-constituents 
at  that  concentration,  the  following  algebraic  expression  for  the  equiva- 
lent conductance  A  of  a  substance  is  obtained: 

A  =  7UC+F  +  7UA-F.  (l) 

For  the  specific  conductance  the  corresponding  expression  is: 

L  =  C7Uc+F  +  C7UA-  F.  (2) 

These  equations  express  the  fundamental  factors  determining 
conductance.  These  factors  are:  (i)  the  number  of  equivalents  of 
(free)  cations  and  anions  (the  carriers  of  the  electricity)  between  the 
electrodes;  (2)  the  mobilities  of  these  ions  (that  is,  their  velocities 
under  a  potential-gradient  of  one  volt  per  centimeter),  and  (3)  the 
quantity  of  electricity  (^6,^oo_coulombs)  carried  by  each  equivalent 
of  ions. 

Any  variation  in  the  equivalent  conductance  of  any  definite  sub- 
stance must  therefore  be  due  to  a  change  either  in  the  number  of  ions 
resulting  from  the  ionization  of  the  substance  or  in  the  mobilities  of  its 
ions;  or  to  both  these  effects.  Thus,  since  the  limiting  value  AO  of 
the  equivalent  conductance  approached  as  the  concentration  ap- 
proaches zero  obviously  corresponds  to  complete  ionization  of  the 
substance  and  to  the  mobility  of  its  ions  in  pure  water,  the  smaller 
values  of  A  observed  at  larger  concentrations  must  arise  from  a  de- 
crease either  in  the  ionization  or  in  the  mobility  of  the  ions  or  from 
changes  in  both  of  these  quantities.  In  other  words,  the  ratio  of  the 
equivalent  conductance  at  any  concentration  to  that  at  zero  concen- 
tration, hereafter  called  simply  the  conductance-ratio,  is  expressed  in 
terms  of  the  fundamental  factors  determining  conduction  by  the 
following  equation: 


This  relation  presupposes  that  the  nature  of  the  ions  does  not 
change  with  increasing  concentration;  thus  the  conductance-ratio 


n8  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

would  not  be  determined  solely  by  the  ionization  of  H2S04  into  H+ 
and  SO4  ions  and  by  the  mobilities  of  these  ions,  if  an  appreciable 
proportion  of  HSO4~  ions  were  present  at  the  larger  concentrations. 
In  such  a  case,  the  conductance-ratio  would  have  no  simple  significance. 

So  long  as  the  solution  is  so  dilute  that  it  does  not  differ  appreciably 
from  pure  water  as  a  medium  offering  frictional  resistance  to  the 
motion  of  the  ions,  the  mobility  of  the  ions  may  be  expected  to  remain 
constant.  In  that  case,  and  provided  that  only  the  same  kinds  of 
ions  are  present  at  the  higher  concentration  as  in  the  very  dilute 
solution,  the  conductance-ratio  is  equal  to  the  ionization;  that  is: 

A  ,  , 

AT7'  (4) 

This  principle  may  be  employed  for  determining  the  ionization  of 
slightly  and  moderately  ionized  substances  up  to  concentrations  such 
as  o.i  to  0.2  normal. 

Prob.  2  p.  — Determination  of  Ionization  from  the  Conductance-Ratio.  — 
From  the  data  given  in  the  table  of  Art.  74,  calculate  the  ionization  of 
acetic  acid  at  18°  in  o.i  and  o.oi  normal  solutions. 

In  moderately  concentrated  solutions  the  frictional  resistance  to 
the  motion  of  the  ions  must  be  appreciably  different  from  that  in 
pure  water;  and  the  conductance-ratio  A/A0  cannot  therefore  be  an 
exact  measure  of  the  ionization.  An  approximate  estimate  of  the 
variation  of  this  frictional  resistance  is  given  by  the  ratio  of  the  vis- 
cosity 77  of  the  solution  to  the  viscosity  775  of  pure  water  at  the  same 
temperature.  This  viscosity-ratio  77/770  is  determined  by  measuring 
the  relative  times  required  for  equal  volumes  of  the  solution  and  of 
pure  water  to  flow  through  the  same  capillary  tube,  when  subjected 
to  the  same  pressure.  A  film  of  the  liquid  adheres  firmly  to  the  walls 
of  the  tube;  and  the  phenomenon  of  the  flow  consists  essentially  in  the 
slipping  of  the  successive  cylindrical  shells  of  liquid  past  one  another. 
Viscosity  is  therefore  a  property  which  depends  on  the  frictional  re- 
sistance to  the  motion  of  the  molecules  of  the  liquid  past  one  another; 
and  it  may  be  expected  to  be  roughly  proportional  to  the  frictional 
resistance  to  the  motion  of  the  ions  through  the  same  solution.  The 
mobilities  of  the  ions  would  then  decrease  in  the  same  proportion  as 
the  viscosity  of  the  solution  increases;  that  is: 

UC+  +  UA-         _  170  /N 

(Ucf)o+(uA-)o        1?' 

By  substituting  this  viscosity-ratio  for  the  ratio  of  the  mobilities  of 


ELECTRICAL  CONDUCTANCE  119 

the  ions  in  the  general  expression  for  the  conductance-ratio  (equation 
(3),  there  results  the  relation 

Ar; 
7  =  -r1-  (6 

Aor/o 

That  is,  in  case  the  viscosity  is  the  only  influence  affecting  the  mobili- 
ties, as  seems  to  be  approx:mately  true  as  long  as  the  concentration 
of  ions  in  the  solution  is  small  (not  greater  than  about  o.oi  normal), 
the  ionization  is  approximately  equal  to  the  conductance-viscosity 
ratio.  This  ratio  therefore  serves  to  determine  the  ionization  of 
slightly  ionized  substances  up  to  fairly  large  concentrat'ons 

Prob.  30.  — Determination  of  Ionization  from  the  Condwtance-Viscosity 
Ratio.  —  Calculate  the  ionization  of  acetic  acid  in  i  normal  solution 
from  the  conductance  values  given  in  the  table  of  Art.  74  and  from  the 
fact  that  the  viscosity  of  i  normal  acetic  acid  solution  at  1 8°  is  1.113 
times  as  great  as  that  of  pure  water. 

The  viscosity  correction  may  be  large  also  in  the  case  of  salt  solu- 
tions, as  is  shown  by  the  following  values  of  the  viscosity-ratio  at  iS° 
and  i  normal. 

Salt  KC1         LiCl         MgCl2       K2S04       MgSO4 

T\IT\§        0.982         1.150        1.213         i.ioi         1.381 

The  deviation  of  the  ratio  from  unity  at  smaller  concentrations  is 
approximately  proportional  to  the  concentration;  thus  at  o.i  normal 
it  is  about  one  tenth  of  that  at  i  normal. 

In  the  case  of  salts,  however,  this  influence  of  viscosity  is  super- 
posed on  another  influence  on  the  mobility  of  the  ions  which  seems  to 
be  large  at  small  ion-concentrations  (even  at  o.oi  normal).  Direct 
evidence  that  there  is  such  an  influence  is  afforded  by  the  change 
in  transference-numbers  at  small  concentrations  (below  0.2  normal) 
discussed  in  Art.  70.  This  influence  makes  the  conductance-ratio 
an  uncertain  measure  of  the  ionization  of  largely  ionized  substances, 
even  at  small  concentrations.  The  ionization  of  such  substances  is 
considered  in  detail  in  Art.  79. 

76.  Conductance  of  the  Separate  Ion-Constituents.  —  Sine*  the 
equivalent  conductance  of  a  substance  at  any  definite  concentBtion 
is  the  sum  of  the  equivalent  conductances  of  its  ion-constituents,  and 
since  the  transference-number  of  either  ion-constituent  is  the  ratio  of 
its  conductance  to  the  total  conductance  (Art.  73),  the  equivalent 
conductance  of  either  ion-constituent  at  that  concentration  is  obtained 


120  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

by  multiplying  its  transference-number  by  the  equivalent  conductance 
of  the  salt. 

The  equivalent  conductance  of  ion-constituents  derived  in  this  way 
increases  with  decreasing  concentration,  and  approaches  a  limiting 
value  as  the  concentration  approaches  zero,  which  is  the  same  what- 
ever be  the  other  ion-constituent  which  is  present  with  it.  This 
limiting  value  is  the  equivalent  conductance  of  the  ion  itself  in  pure 
water,  hereafter  called  simply  the  ion-conductance. 

The  ion-conductances  for  the  various  ions  have  been  derived,  as 
illustrated  by  the  following  problem,  by  the  additivity  principle  from 
the  equivalent  conductances  of  their  salts  at  zero  concentration  with 
the  aid  of  a  single  transference-number,  that  for  potassium  chloride, 
which  is  the  one  best  known  at  small  concentrations. 

Pr ob.  jr.  —  Determination  of  Ion-Conductances.  —  a.  Calculate  the 
ion-conductances  of  the  potassium  and  chloride  ions  at  18°  from  the 
transference  values  given  in  Art.  70  and  from  the  value  129.6  of  the 
equivalent  conductance  at  zero  concentration  of  potassium  chloride. 
b.  Calculate  the  ion-conductances  of  the  sodium,  hydrogen,  and  sulfate 
ions  at  18°  from  the  conductance  values  in  the  table  in  Art.  74. 

The  following  table  contains  the  values  of  the  ion-conductances  for 
some  important  ions  at  18°  and  25°. 


VALUES  OP  THE  ION-CONDUCTANCES 

Anions  18°  25° 

OH-  173.  195. 

F-  46.6  54.3 

Cl-  65.3  75-2 

Br~  67.4  77.5 

I~  66.4  76.3 

SON"  56.5 

NO3~  61.7  70.6 

C1O3-  54-9  63.2 

BrO3~  47-7  54-9 

I03-  33-8  39-3 

C2H3Oj  35-  41. 

SO4=  67.7  78.4 

CrO4=  72 

€204-  61.  — 


Cd+  + 


ELECTRICAL  CONDUCTANCE  121 

The  ion-conductance  always  increases  rapidly  with  the  temperature; 
for  example,  at  18°  by  about  1.8  percent  per  degree  in  the  case  of  the 
fast-moving  hydrogen  and  hydroxide  ions  and  by  2.1  to  2.6  percent  in 
the  case  of  other  ions.  This  increase  of  conductance  evidently  cor- 
responds to  an  equal  percentage  increase  in  the  mobility  of  the  ion. 

Prob.  32.  — Ion-Conductance  in  Relation  to  Atomic  Weight  and  Hydra- 
tion  of  the  Ions.  —  a.  From  a  consideration  of  the  table  state  the  relation 
that  exists  between  the  ion-conductances  and  the  atomic  weights  of  the 
different  elementary  ions  belonging  to  the  alkali  group  and  to  the  al- 
kaline-earth group,  b.  State  how  this  relation  might  be  explained  by 
^fc^reference  to  the  hydration  of  the  ions  described  at  the  end  of  Art.  71. 

From  these  ion-conductances  the  A0  value  for  the  various  salts 
can  be  obtained  by  simple  addition.  This  fact  is  of  especial  impor- 
tance in  the  case  of  substances  for  which  the  AO  value  cannot  be 
derived  from  conductance  measurements  by  extrapolation.  This  is 
true  of  weak  bases  and  acids,  such  as  ammonium  hydroxide  and  acetic 
acid,  whose  ionization  is  far  from  complete  even  in  dilute  solution; 
it  is  also  true  of  salts,  such  as  ammonium  acetate,  which  are  appreci- 
ably hydrolyzed  in  dilute  solution. 

77.  Determination  of  the  Concentration  of  Ions  and  of  Largely 
Ionized  Substances  by  Conductance  Measurements.  —  Conductance 
measurements  are  often  employed,  both  in  scientific  and  technical 
work,  for  determining  the  concentration  of  the  ions  in  solutions  and 
deriving  therefrom  the  concentration  of  the  largely  ionized  substances 
that  may  be  present 

When  the  ion  concentration  is  so  small  (not  greater,  for  example, 
than  o.ooi  normal)  that  its  influence  in  decreasing  the  mobilities 
of  the  ions  can  be  neglected  without  serious  error,  the  ion-con- 
centration can  be  calculated  from  the  measured  specific  conductance 
simply  with  the  aid  of  the  ion-conductances  (as  in  Probs.  33  and  34). 
When  the  ion-concentration  is  so  large  that  this  influence  cannot  be 
neglected,  it  is  first  roughly  estimated  with  the  aid  of  the  ion-con- 
ductances, and  the  concentration  of  the  substance  is  then  accurately 
determined  by  using  the  known  values  of  the  equivalent  conductance 
of  the  substance  in  the  neighborhood  of  the  estimated  concentration 
(as  in  Prob.  35). 

Prob.  33.  —  Determination  of  the  Solubility  of  Slightly  Soluble  Sub-       , 
stances.  —  When  water  at  18°  is  saturated  with  silver  chloride,  its      f^ 
specific  conductance  is  increased  by  i .  2 5  X  io~6  reciprocal  ohm.     Calcu- 
late the  solubility  of  silver  chloride  in  equivalents  per  liter. 


122  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

Prob.  34.  —  Determination  of  the  Concentration  of  the  Ions  in  Pure 
Water.  —  The  specific  conductance  of  pure  water,  resulting  from  its 
dissociation  into  H+  and  OH~  ions,  is  3.8Xio~8  reciprocal  ohm  at  18°, 
and  5.8Xio~8  reciprocal  ohm  at  25°.  Calculate  the  concentration 
of  the  ions  in  equivalents  per  liter  at  each  of  these  temperatures. 

Prob.  55.  —  Determination  of  the  Hydrolysis  of  Salts.  —  A  0.02  normal 
NH4CN  solution  has  at  25°  a  specific  conductance  of  0.00136  reciprocal 
ohms.  This  salt  is  largely  hydrolyzed  in  accordance  with  the  equation 
NH4CN+H2O  =  NH4OH+HCN,  and  the  NH4OH  and  HCN  are  so 
slightly  ionized  in  the  presence  of  their  salt  as  to  have  only  a  negligible 
conductance.  The  ion-conductances  of  NH4+  and  CN~  at  25°  are 
74.5  and  70,  respectively.  Assuming  that  the  equivalent  conductance 
of  unhydrolyzed  NH4CN  decreases  with  the  concentration  in  the  same 
ratio  as  does  that  of  NaCl  (given  in  the  table  of  Art.  74)  calculate  the 
fraction  of  the  salt  that  is  hydrolyzed.  Ans.  50%. 


THE  IONIZATION  OF  SUBSTANCES  123 

V.    THE   IONIZATION  OF   SUBSTANCES 

78.  lonization  of  Substances  Not  Largely  Ionized.  —  As  shown  in 
Art.  75,  the  conductance-ratio  or  conductance- viscosity  ratio  can  be 
considered  substantially  equal  to  the  ionization  so  long  as  the  total 
concentration  of  ions  in  the  solution  is  small  (not  greater  than  about 
o.oi  normal).     The  ionization  values  for  many  slightly  and  moder- 
ately ionized  acids  and  bases  have  been  determined  hi  this  way,  with 
the  results  stated  in  the  following  paragraph. 

Acids  and  bases,  unlike  salts,  exhibit  at  any  moderate  concentra- 
tion, such  as  o.i  normal,  every  possible  degree  of  ionization  between 
a  small  fraction  of  one  percent  and  nearly  100  percent.  There  is,  to 
be  sure,  a  fairly  large  group  of  acids  and  bases,  including  HC1,  HBr, 
HI,  HNO3,  HC1O3,  KOH,  NaOH,  LiOH,  Ba(OH)2,  Ca(OH)2,  which 
like  neutral  salts  are  almost  completely  ionized.  But  outside  of  this 
group  all  possible  values  are  met  with,  as  illustrated  by  the  following 
approximate  values  of  the  percentage  ionization  (100  7)  at  25°  and 
o.i  normal. 

PERCENTAGE  IONIZATION  OF  ACIDS  AND  BASES 

H2S03  (into  H+ and  HSO3~) 34% 

H3PO4  (into  H+  and  H2PO4-) 28 

HNO2andHF 8 

_HC2H3O2  and  NH4OH i 

H2S,  H2CO3,  and  HC1O o.i 

HCNandHBO2         ........  0.002 

Polybasic  acids  are  known  to  ionize  in  stages,  giving  rise  to  the 
intermediate  ion;  and  the  first  hydrogen  is  almost  always  much  more 
dissociated  than  the  second,  and  the  second  much  more  than  the  third 
Thus  H2SO3  at  o.i  normal  at  25°  is  about  34  percent  dissociated  into 
H+  and  HSO3~,  and  less  than  o.i  per  cent  dissociated  into  H+  and 
SO3  ~ .  Methods  by  which  the  dissociation  of  the  successive  hydrogens 
can  be  determined  are  referred  to  in  Art.  106. 

A  few  salts  of  higher  valence  types  form  marked  exceptions  to  the 
principle  that  salts  are  almost  completely  ionized.  Thus  at  o.i 
normal  and  25°,  the  value  of  the  conductance-ratio  for  cadmium 
chloride  is  0.45  and  the  values  for  the  mercuric  halides  are  less  than 

O.OOI. 

79.  lonization  of  Largely  Ionized  Substances.  —  There  are  many 
facts  which  indicate  that  neutral  salts  and  largely  ionized  univalent 


124  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

acids  and  bases,  like  hydrochloric  acid  and  potassium  hydroxide,  are 
completely  or  almost  completely  ionized  up  to  moderate  concentra- 
tions, such  as  0.2  normal,  and  that  the  decrease  in  the  equivalent 
conductance  of  such  substances  up  to  these  concentrations  is  therefore 
due  wholly  or  almost  wholly  to  a  decrease  in  the  mobility  of  the  ions. 
This  hypothesis  of  complete  ionization  is  in  contrast  with  the  hypothe- 
sis of  constant  ion-mobility,  according  to  which  the  decrease  in  equiva- 
lent conductance  is  wholly  or  mainly  due  to  a  decrease  in  the  ioniza- 
tion of  the  substance. 

The  hypothesis  of  constant  ion-mobility,  which  has  ordinarily  been 
employed  in  the  development  of  the  ionic  theory,  is  directly  contra- 
dicted by  the  experimental  evidence,  mentioned  in  Art.  70,  which 
indicates  that  the  transference-number  for  certain  substances,  and 
therefore  the  ratio  of  the  mobilities  of  their  two  ions,  changes  con- 
siderably with  the  concentration,  even  when  this  is  not  greater  than 
o.i  normal. 

The  hypothesis  of  constant  ion-mobility,  moreover,  leads  to  a  re- 
markable uniformity  in  the  ionization-values  for  substances  of  the 
same  valence  type.  Namely,  if  the  conductance-ratio  or  the  con- 
ductance-viscosity ratio  is  regarded  as  a  true  measure  of  ionization, 
it  is  necessary  to  conclude  from  the  facts  (stated  in  Art.  74)  in  regard 
to  the  fractional  decrease  of  equivalent  conductance  with  the  con- 
centration that  different  salts  of  the  same  valence  type  have  at  any 
definite  concentration  nearly  equal  ionization  values.  Thus  at  18° 
and  o.i  normal  the  conductance- viscosity  ratio,  and  under  this  hy- 
pothesis the  ionization,  has  for  various  salts  the  following  values: 
0.852  for  NaCl  0.814  for  AgN03  0.827  for  KC1O3 

0.759  for  BaCl2  0.731  for  Ca(NOs)2  0.704  for  Na£SO4 
This  behavior  of  salts  would  indicate  that  the  extent  to  which  their 
ions  in  solution  combine  to  form  neutral  molecules  is  primarily  deter- 
mined, not  by  their  chemical  nature,  but  only  by  the  magnitude  of 
the  charges  which  they  carry.  While  this  is  possible,  it  would  signify 
a  new  type  of  chemical  combination;  for  all  other  chemical  reactions 
are  found  to  depend  in  a  highly  specific  way  on  the  nature  of  the 
atoms  and  atom-groups  uniting  with  one  another,  as  is  the  case  even 
in  the  union  of  hydrogen-ions  with  anions  to  form  acids  or  of  hydroxide- 
ions  with  cations  to  form  bases. 

Another  phenomenon,  which  seems  anomalous  if  the  conductance- 
ratio  is  regarded  as  equal  to  the  ionization,  is  the  approximate  pro- 


THE  IONIZATION  OF  SUBSTANCES  125 

portionality  between  the  decrease  of  this  ratio  and  the  cube-root  of 
the  concentration,  whether  the  salt  be  of  the  imiunivalent  type  or  the 
unibivalent  type;  whereas  the  mass-action  law  requires  (as  shown 
in  Art.  103)  for  either  of  these  types  a  relation  very  different  from  the 
cube-root  function,  and  moreover,  a  relation  which  is  a  different 
function  for  the  two  valence  types. 

This  divergence  which  the  ionization  values  derived  under  the 
hypothesis  of  constant  ion-mobility  show  from  the  requirements  of  the 
mass-action  law  can,  to  be  sure,  be  explained  by  assuming  that  the 
ions  do  not  behave  as  perfect  solutes  and  that  therefore  they  exert 
an  abnormal  mass-action  effect.  For,  as  will  be  shown  in  Art.  141, 
the  mass-action  law  is  valid  only  when  the  substances  involved  in  the 
equilibrium  to  which  it  is  applied  (in  this  case  to  the  equilibrium 
between  the  ions  and  the  unionized  part  of  the  salt)  behave  as  perfect 
solutes.  Moreover,  the  ionization  values  derived  from  the  effect  of 
salts  on  the  molal  properties  of  solutions,  such  as  the  vapor-pressure 
and  freezing-point,  under  the  assumption  that  the  ions  and  unionized 
molecules  are  perfect  solutes,  show  that  this  assumption  cannot  be 
correct;  for,  though  these  ionization  values  as  a  rule  agree  well  with 
those  derived  from  the  conductance,  yet  like  the  latter  they  do  not 
change  with  the  concentration  even  approximately  in  accordance 
with  the  mass-action  law. 

The  foregoing  considerations  may  be  summed  up  as  follows: 
(i)  either  hypothesis  explains  equally  well  the  conduction  through  solu- 
tions; (2)  either  hypothesis,  in  order  to  account  for  all  the  facts,  must 
be  supplemented  by  the  further  hypothesis  that  the  ions  are  not 
perfect  solutes  and  therefore  have  abnormal  effects  on  the  molal 
properties  of  solutions  and  an  abnormal  mass-action  in  determining 
equilibrium;  (3)  the  hypothesis  of  constant  ion-mobility  leads  to 
improbable  consequences  in  regard  to  the  ionization  of  largely  ionized 
substances,  and  requires  for  the  expression  of  the  mass-action  of  an 
ion-constituent  at  each  concentration,  not  only  (as  does  the  other 
hypothesis)  a  coefficient  expressing  the  deviation  of  the  behavior  of 
the  ions  from  that  of  perfect  solutes,  but  also  an  ionization-value 
expressing  their  concentration  which  cannot  be  determined  by  any 
reliable  method. 

Since  in  the  present  state  of  our  knowledge  the  hypothesis  that  the 
ionization  of  largely  ionized  substances  is  substantially  complete  up 
to  moderate  concentrations  serves  to  explain  the  facts  as  satisfactorily 


1 26  ELECTROLYTIC  BEHAVIOR  OF  SOLUTIONS 

as  the  alternative  hypothesis,  and  since  it  is  simpler  in  its  application, 
it  is  employed  in  the  following  chapters.  In  using  this  principle  the 
possibility  must,  however,  be  kept  in  mind  that  a  more  complete 
study  of  the  properties  of  largely  ionized  substances  may  show  that 
neither  of  these  hypotheses  is  alone  adequate  to  explain  fully  their 
behavior;  in  other  words,  it  may  prove  necessary  to  assume  that  both 
the  ion-mobilities  and  the  ionization  of  these  substances  vary  with  the 
concentration. 


CONDUCTANCE  AND  TRANSFERENCE  127 

VI.    APPLICATIONS  OF  THE  LAWS  OF  CONDUCTANCE  AND  TRANSFERENCE 

80.  Review  Problems.      ^ 

Prob.  36.  —  0.  A  0.06  formal  solution  of  C12  in  water  in  which  a  partial 
reaction  has  taken  place  according  to  the  equation  Cl2-fH2O=HCl 
+  HC1O  has  a  freezing-point  of  —  0.180°.  Calculate  what  fraction  of 
the  C12  has  undergone  change,  regarding  the  HC1O  as  unionized  and 
the  ions  of  the  HC1  as  perfect  solutes,  b.  Calculate  the  fraction  that 
has  undergone  change  from  the  following  data.  The  solution  has  a 
specific  conductance  of  0.0x5472  reciprocal  ohm  at  o°.  The  equivalent 
conductances  of  HC1  at  o°  between  o.o  and  o.i  normal  are  0.700  times 
those  at  18°  given  in  Art.  74.  Ans.  a,  0.31. 

Prob.  37.  — A  o.i  normal  solution  of  chlorplatinic  acid,  H2PtCl6,  is 
electrolyzed  at  25°,  platinum  alone  being  deposited  on  the  cathode. 
Assume  that  this  acid  dissociates  into  2H+  and  PtCl6=  ions,  whose 
equivalent  conductances  at  25°  are  347  and  68,  respectively,  and  that 
these  ions  are  the  only  ones  present  that  carry  a  significant  fraction  of 
the  current  through,  the  solution,  a.  Write  the  electrochemical  equa- 
tion expressing  the  cathode  process,  b.  Find  what  changes  in  the 
hydrogen,  chlorine,  and  platinum  content,  expressed  in  terms  of  the 
number  of  atomic  weights  of  each  element,  take  place  in  the  cathode 
portion  when  4825  coulombs  are  passed.  Ans.  b,  loss  of  0.0166  at.  wt. 
of  Pt. 

*Prob.  38.  —  From  the  fact  that  in  a  moving-boundary  experiment, 
like  that  represented  in  the  figure  of  Art.  69,  the  ion-constituent  C' 
must  move  in  the  region  behind  the  boundary  with  the  same  actual 
velocity  as  the  ion-constituent  C  moves  in  the  region  beyond  the 
boundary,  and  from  the  fact  that  the  respective  potential-gradients 
in  these  two  regions  are  proportional  to  the  specific  resistances  there 
prevailing,  show  that  the  ratio  of  the  resulting  concentrations  of  the 
salts  C'A  and  CA  in  these  two  regions  is  equal  to  the  ratio  of  the  trans- 
ference-number of  the  ion-constituent  C'  in  the  solution  of  C'A  to  that 
of  the  ion-constituent  C  in  the  solution  of  CA. 


PART  II 

THE   RATE   AND    EQUILIBRIUM    OF    CHEMICAL 

CHANGES  FROM   MASS-ACTION  AND 

PHASE  VIEW-POINTS 


CHAPTER  VI 
THE   RATE   OF   CHEMICAL   CHANGES 


I.  THE  EFFECT  OF  CONCENTRATION  ON  REACTIONS  BETWEEN 
SOLUTES  OR  GASEOUS  SUBSTANCES 

81.  Concept  of  Reaction-rate.  —  The  reaction-rate  dc/dt  of  a  chem- 
ical change  taking  place  between  dissolved  substances  may  be  denned 
to  be  the  increase  dc  either  in  the  weight-normality  or  in  the  normal 
concentration  (Art.  35)  of  the  reaction  products  in  an  infinitesimal 
time  dt  divided  by  that  time.    This  concept  is  illustrated  by  the  fol- 
lowing problem. 

Prob.  i.  —  The  Concept  of  Reaction-Rate.  —  Sodium  hydroxide  and 
methyl  acetate  in  dilute  aqueous  solution  react  with  each  other,  as 
expressed  by  the  equation: 

CH3C2H302  +NaOH  =  CH3OH +NaC2H3O2. 

Starting  with  a  solution  o.oi  weight-normal  (wn.)  in  methyl  acetate 
and  sodium  hydroxide  and  keeping  the  temperature  constant  at  25°, 
the  concentrations  of  the  methyl  alcohol  and  sodium  acetate  (in  milli- 
equivalents  per  1000  g.  of  water)  after  various  times  (in  minutes)  are 
found  to  be  as  follows: 

Time 02          5          10        20        40        8 

Concentration  .     o    i;gi     3.70     5.41      7.02     8.25     10.00 

a.  Plot  these  concentrations  as  ordinates  against  the  times  as  abscissas. 

b.  Determine  from  the  plot  the  reaction-rate  at  the  start,  after  10 
minutes,  and  after  30  minutes,  stating  the  units  in  which  the  reaction- 
rate  is  expressed,     c.  Suggest  an  analytical  method  by  which  the  rate 
of  this  reaction  could  be  followed  experimentally. 

82.  The  Mass-Action  Law  of  Reaction-Rate  between  Solutes.  — 
The  effect  of  concentration  on  the  rate  of  chemical  changes  between 
solutes  is  expressed  by  the  following  principle,  known  as  the  mass- 
action  law  of  reaction-rate.    The  rate  at  any  definite  moment  of  a 
chemical  reaction  which  takes  place  at  a  constant  temperature  be- 
tween perfect  solutes  completely  in  one  direction  is  proportional 
to  the  concentration,  expressed  either  as  weight-normality  or  normal 
concentration,  of  each  of  the  reacting  substances  at  that  moment. 
That  is,  for  the  general  case  that  any  substances  A,  B,  C,...  react 
with  one  another: 


132  THE  RATE  OF  CHEMICAL  CHANGES 

The  proportionality-constant  k  which  occurs  in  this  equation  is 
called  the  specific  reaction-rate.  It  evidently  represents  the  rate  which 
the  reaction  would  have,  under  the  assumption  of  proportionality, 
if  the  weight-normality  or  the  normal  concentration  of  each  of  the 
reacting  substances  were  unity. 

This  law  is  exact  for  perfect  solutes;  but,  like  other  laws  of  perfect 
solutions,  it  holds  true  approximately  up  to  moderate  concentrations. 
The  statement  of  the  law  here  given  is  an  incomplete  one  applicable 
only  to  reactions  involving  a  single  molecule  of  each  of  the  reacting 
substances.  A  general  statement  of  the  law  will  be  made  after  the 
mechanism  of  reactions  has  been  considered. 

83.  First-Order  Reactions.  —  The  expression  for  the  reaction- 
rate  is  simplest  in  the  case  of  reactions  in  which  only  one  substance 
undergoes  a  change  in  concentration.  The  rate  in  this  case  is  ex- 
pressed by  a  differential  equation  of  the  first  degree;  and  such  reactions 
are  called  first-order  reactions  or  unimolecular  reactions. 

An  example  of  a  first-order  reaction  is  the  hydrolysis  (splitting  by 
water)  of  cane-sugar,  which  takes  place  in  dilute  aqueous  solution 
in  the  presence  of  acid  according  to  the  equation: 

Ci2H22On  (cane-sugar)  +  H2O  =  C6Hi206  (glucose)  +  C6Hi2O6(fructose). 

Although  two  substances  take  part  in  this  reaction,  only  one  of  them, 
the  cane-sugar,  need  be  considered;  for  the  mass-action  or  concentra- 
tion-effect of  the  water,  being  present  in  large  excess,  does  not  change 
appreciably  during  the  course  of  the  reaction.  The  acid  acts  catalyti- 
cally,  accelerating  the  reaction  without  being  consumed  by  it.  In 
formulating  the  reaction-rate  equation  and  in  evaluating  the  specific 
reaction-rate,  it  is  usual  to  leave  out  of  account  the  concentrations  of 
substances  which,  like  these,  do  not  change  with  the  progress  of  the 
reaction. 

Prob.  2.  —  Formulation  of  the  Equations  for  First-Order  Reactions.  — 
a.  Formulate  the  differential  equation  expressing  the  rate  at  which 
the  hydrolysis  of  cane-sugar  takes  place,  representing  by  C0  the  con- 
centration of  the  cane-sugar  at  the  start,  and  by  c  the  concentration 
of  the  glucose  or  fructose  which  has  resulted  at  the  time  t.  b.  Integrate 
this  equation  between  the  time  limits,  o  and  /,  and  the  corresponding 
limits  of  the  concentration. 

Prob.  3.  —  Applications  of  the  First-Order  Equations.  —  In  a  solution 
containing  0.3  mol  of  cane-sugar  and  o.i  mol  of  HC1  in  1000  g.  of  water, 
it  is  found  (by  measuring  with  a  polarimeter  the  change  in  the  optical 
rotatory  power)  that  32%  of  the  sugar  is  hydrolyzed  at  48°  in  20  min- 


THE  EFFECT  OF  CONCENTRATION  133 

utes.  a.  Calculate  the  specific  reaction-rate  and  the  actual  rates  at 
the  beginning  and  at  the  expiration  of  20  minutes,  b.  Calculate  the 
percentage  of  sugar  hydrolyzed  after  40  minutes. 

84.  Second-Order  and  Third-Order  Reactions.  —  Reactions  whose 
rates  are  proportional  to  the  concentrations  of  each  of  two  substances 
or  of  each  of  three  substances,  are  called  second-order  or  third-order 
reactions,  or  bimolecular  or  trimolecular  reactions.  And  in  general, 
the  order  of  a  reaction  is  said  to  be  equal  to  the  degree  of  the  differential 
equation  expressing  its  rate. 

Formulation  and  Application  of  the  Equations  for  Second-Order  Re- 
actions. — 

Prob.  4.  —  a.  Formulate  the  differential  equation  expressing  the  rate 
in  dilute  aqueous  solution  of  the  reaction  between  methyl  acetate  and 
sodium  hydroxide,  in  terms  of  the  initial  concentrations  cAo  and  CBO 
of  these  two  substances,  and  the  concentration  c  of  the  reaction-products 
at  the  time  t.  b.  Integrate  this  equation  between  the  time  limits  o  and  / 
for  the  case  that  the  initial  concentrations  of  the  two  substances  have 
the  same  value  CQ.  c.  Integrate  the  equation  (with  the  aid  of  a  table 
of  integrals,  if  preferred)  also  for  the  case  that  the  initial  concentrations 
CAO  and  cBo  are  different  from  each  other. 

Prob  5.  —  When  o.oi  mol  of  methyl  acetate  and  o.oi  mol  of  NaOH 
are  dissolved  at  25°  in  1000  g.  of  water,  n.8%  of  the  ester  is  decom- 
posed per  minute  at  the  start,  a.  Calculate  the  time  required  for 
one  half  of  the  ester  in  this  mixture  to  be  decomposed,  b.  Calculate 
the  time  required  for  this  decomposition  when  o.oi  mol  of  methyl 
acetate  and  0.02  mol  of  NaOH  are  dissolved  in  1000  g.  of  water.  Ans. 
b,  3.4  min. 

The  expressions  for  a  reaction  of  a  third  order  (between  three  sub- 
stances A,  B,  and  C)  can  be  similarly  derived. 

For  the  case  that  all  the  substances  involved  have  the  same  initial 
concentration  c0  the  integrated  expressions  for  the  three  orders  are: 

log-^-  =  £/;  — *-  =  kt;  and   — ^--^  =  2*/. 

&c0-c  CQ-C      c0  (c0— c)2    c02 

The  following  expressions  for  the  fraction  x  of  the  reacting  substances 
transformed  can  be  derived  from  these  integrated  expressions  for 
reactions  of  the  first,  second,  and  third  orders: 


log-   -  =  kt--    -  =  ktc0;  and^-£=2^Co2. 

O    T  /y  7T  /v*  (T  X) 

X          *-v  X          *v  \ x          *v/ 

Derivation  and  Application  of  the  Expressions  for  the  Fraction  of  the 
Reacting  Substances  Transformed.  — 

pro0t  6.  —  Derive  the  expressions  for  the  fraction  transformed  given 
in  the  preceding  text. 


134  THE  RATE  OF  CHEMICAL  CHANGES 

Prob.  7.  —  At  25°  the  specific  rate  of  the  reaction  between  sodium 
hydroxide  and  methyl  acetate  is  1.8  times  as  great  as  that  of  the  reaction 
between  sodium  hydroxide  and  ethyl  acetate.  Find  the  ratio  of  the 
times  required  for  decomposing  90%  of  the  two  esters  when  equivalent 
quantities  are  present  at  the  start. 

Prob.  8.  —  Show  from  the  above  expressions  how  the  time  required 
for  the  transformation  of  any  definite  fraction  of  a  definite  quantity  of 
the  reacting  substances  would  vary  if  the  quantity  of  solvent  in  which 
they  are  dissolved  were  increased  tenfold,  a,  in  the  case  of  the  cane- 
sugar  hydrolysis  (Prob.  3)  ;  b,  in  the  case  of  the  decomposition  of  methyl 
acetate  by  sodium  hydroxide  (Prob.  5). 

85.   The  Mechanism  of  Reactions  and  the  General  Mass-Action 
Law  of  Reaction-rate.  —  The  rate  of  certain  reactions,  such  as, 
2FeCl3  +  SnCl?  =  2FeCl2  +  SnCl4,    and 
2  AgC2H3O2  +  NaCHO2  =  2  Ag  +  CO2  +  HC2H3O2  +  NaC2H3O2, 
in  which  two  molecules  of  one  of  the  substances  are  involved,  has  been 
found  to  be  proportional  to  the  square  (instead  of  to  the  first  power) 
of  the  concentration  of  that  substance.     In  other  words,  it  is  found 
that  these  reactions  are  of  the  third  order,  instead  of  the  second  order. 
This  justifies  the  conclusion  that  the  number  of  reacting  molecules 
determines  the  law  of  the  rate,  and  this  conclusion  is  further  substanti- 
ated by  the  fact  that  it  harmonizes  the  mass-action  law  of  reaction- 
rate  with  the  mass-action  law  of  chemical  equilibrium  (see  Art.  98). 

The  provisional  statement  (Art.  82)  of  the  law  of  reaction-rate 
should  therefore  be  modified  so  as  to  state  that  the  rate  is  propor- 
tional to  the  concentration  of  each  of  the  reacting  substances  raised 
to  a  power  equal  to  the  number  of  its  molecules  which  interact  with 
the  molecules  of  the  other  substances  in  the  molecular  process  on 
which  the  occurrence  of  the  reaction  depends.  Thus,  in  the  case  of  a 
reaction  aA  +  frB+cC.  .  ,  =  eE  +  .  .  .,  whose  occurrence  requires  the 
interaction  of  a  molecules  of  Aj  b  molecules  of  B,  c  molecules  of  C,  etc., 
the  mass-action  law  of  reaction-rate  is  expressed  by  the  equation: 


Prob.  g.  —  Application  of  the  Law  of  Reaction-Rate.  —  Equal  volumes 
of  0.2  wn.  solutions  of  silver  acetate  (AgC2H3O2)  and  sodium  formate 
(NaCHOa)  are  mixed  at  100°  ;  and  after  definite  intervals  of  time  samples 
are  removed,  and  the  undecomposed  silver  acetate  is  titrated  with 
potassium  thiocyanate.  Its  weight-normality  is  found  to  be  0.067 
after  2  minutes,  0.047  after  6  minutes,  and  0.032  after  14  minutes. 
Show  from  these  data  that  this  reaction  conforms  more  closely  to  the 
expression  of  the  third  order  than  to  that  of  the  second  order. 


THE  EFFECT  OF  CONCENTRATION 


135 


It  is  found,  however,  that  many  reactions  which  apparently  in- 
volve three  or  more  molecules  conform  to  the  expression  of  the 
second  order.  This  is  probably  to  be  explained  by  the  consideration 
that  the  reaction  expressed  by  the  usual  chemical  equation  takes 
place  in  stages,  and  that  the  stage  which  requires  appreciable  time  is 
a  reaction  between  two  molecules.  Thus  the  second-order  reaction 


may  be  considered  to  take  place  in  the  two  stages, 

H2O2  +  HI=H2O  +  HIO,  and  HIO+HI=H2O+I2; 
the  first  requiring  a  measurable  time,  and  the  second  taking  place 
almost  instantaneously  as  soon  as  any  HIO  is  formed  by  the  first  re- 
action. It  is  therefore  necessary  in  the  case  of  complex  reactions  to 
know  their  mechanism,  that  is,  the  molecular  process  by  which  they 
take  place,  in  order  to  predict  the  law  of  their  rate;  and  conversely, 
the  law  of  the  reaction-rate  throws  light  on  the  mechanism  of  the 
reaction. 

The  Mechanism  of  Reactions.  — 

Prob.  10.  —  Suggest  an  explanation  of  the  fact  that  the  rate  of  the 
reaction  H+BrO3~+6H+I~=H+Br~+3l2+3H2O  is  proportional  to  the 
first  power  of  the  concentration  of  the  BrO3~  and  of  that  of  the  I~~. 

Prob.  ii.  —  In  aqueous  solutions  the  measured  rates  of  the  chemical 
changes  expressed  by  the  following  equations  show  that  both  of  them 
are  second-order  reactions,  and  that  they  have  not  far  from  the  same 
specific  reaction-rates  : 

2CH3C2H302+Ba(OH)2  =  2CH3OH+Ba(C2H3O2)2. 

CH3C2H3O2  +NaOH  =  CH3OH  +NaC2H3O2. 

Suggest  a  mechanism  that  accounts  for  the  fact  that  the  first  of  these 
changes,  like  the  second,  is  a  reaction  of  the  second  order,  and  that  the 
specific  rates  of  the  two  reactions  are  not  far  from  equal;  and  write  an 
equation  expressing  the  actual  reaction  whose  rate  determines  the 
progress  of  the  decomposition  of  the  methyl  acetate. 

Prob.  12.  —  Assuming  the  correctness  of  the  suggested  mechanism 
of  the  methyl  acetate  decomposition,  determine  the  quantity  of  free 
NaOH  present  (because  of  hydrolysis)  in  a  o.i  wn.  NaCN  solution  at 
25°  from  the  fact  that  at  25°  the  initial  rate  of  decomposition  of  methyl 
acetate  is  9.0  times  as  great  in  a  solution  o.oi  wn.  in  NaOH  as  in  a 
solution  o.i  wn.  in  NaCN. 

86.  The  Mass-Action  Law  of  Reaction-Rate  between  Gaseous 
Substances.  —  The  general  mass-action  law  (Art.  85)  is  applicable 
to  the  rate  of  reactions  between  gaseous  substances  in  substantially 
the  same  form  as  to  those  between  solutes.  It  is,  however,  usually 


136  THE  RATE  OF  CHEMICAL  CHANGES 

more  convenient  to  express  the  concentrations  in  terms  of  mols  per 
liter;  but  in  that  case  it  is  necessary  to  refer  the  rate  to  a  specified 
substance.  Thus  the  rate  of  the  reaction  2  A  +  B  =  2E  (for  example, 
2NO  +  Oz  =  2NO2)  at  constant  temperature  may  be  expressed  by  any 
'of  the  equations: 

-^  =  kcfc»-  -^-=kc^c^  -d-^=Uc?c^  where  k'  =  $k. 

For  the  case  that  the  volume  of  the  gas  mixture  remains  constant, 
this  equation  expressed  in  terms  of  the  molal  concentrations  of  the 
reacting  substances  A  and  B  at  the  start  and  the  concentration  of  the 
product  E  at  the  time  t,  becomes: 


It  is  important  to  note  that  many  gas  reactions  are  very  sensitive 
to  catalytic  influences.  In  fact,  it  is  not  uncommon  for  gas  reactions 
to  take  place  to  an  appreciable  extent  only  along  the  walls  of  the  con- 
taining vessel.  The  rate  of  such  catalytic  or  contact  reactions  is  riot 
determined  by  the  mass-action  law,  which  is  applicable  only  to  re- 
actions that  take  place  within  the  gas. 

Rate  of  the  Reactions  in  the  Arc  Process  of  Nitrogen  Fixation.  — 
Prob.  13.  —  In  the  arc  process  of  nitrogen-fixation  air  is  brought  in 
contact  with  an  electric  arc,  the  gas  from  the  arc  is  rapidly  cooled  (by 
admixture  with  cold  air  and  contact  with  the  walls  of  the  furnace),  and 
under  definite  conditions  the  gas  emerges  from  the  furnace  at  1250°  and 
i  atm.  with  a  content  of  2  mol-percent  of  nitric  oxide  (NO).  Deter- 
mine what  percentage  of  this  nitric  oxide  would  be  decomposed  into 
nitrogen  and  oxygen  if  the  gas  was  allowed  to  remain  at  1250°  for  10 
minutes,  from  the  facts  that  at  this  temperature  the  equilibrium  condi- 
tions correspond  to  a  substantially  complete  decomposition,  and  the 
specific  reaction-rate  in  mols  of  NO  destroyed  per  liter  per  minute  is 
50.  Ans.  7.4. 

Prob.  14.  —  When  the  arc-process  gas  containing  2  mol-percent  of 
nitric  oxide  is  cooled  at  i  atm.  to  20°  the  reaction  2NO  +  O2  =  2NO2 
takes  place.  From  the  facts  that  the  specific  reaction-rate  in  mols  of 
NO2  produced  per  liter  per  second  at  20°  is  9400,  and  the  equilibrium 
conditions  correspond  to  substantially  complete  formation  of  NO2, 
find  the  time  that  must  be  allowed  to  convert  90%  of  the  NO  into  NO2. 
(In  integrating  the  differential  equation  regard  as  constant  the  volume 
of  the  mixture  and  the  concentration  of  the  oxygen.)  Ans.  138  sec. 

87.  Simultaneous  Reactions.  —  It  often  happens  that  two  or  more 
related  reactions  are  taking  place  simultaneously.  Some  of  the 
important  types  of  such  reactions  are  the  following. 


THE  EFFECT  OF  CONCENTRATION 

(1)  A  substance  may  be  undergoing  change  in  two  different  senses; 
thus,  in  the  presence  of  i^din^whidi_acts_as_a  catalyst,  chlorbenzene 
and  chlorine  in  carbon  bisulfide  solution  undergo  the  two  independent 
reactions  represented  by  the  equations: 

C6H5C1  +  C12 = HC1  +  ortho  C6H4C12. 
C6H5C1  +  C12 = HC1 + para  C6H4C12. 

(2)  The  products  resulting  from  one  reaction  may  undergo  change 
by  a  second  reaction;  thus,  when  dimethyl  oxalate  and  sodium  hy- 
droxide are  brought  together  in  aqueous  solution,  the  following  reac- 
tions take  place  consecutively: 

(CH3)2C2O4 + NaOH  =  (CH3)NaC2O4  +  CH3OH. 
(CH3)NaC2O4 + NaOH = Na2C204  +  CH3OH. 

(3)  The  products  resulting  from  a  reaction  may  tend  to  react  with 
one  another  in  the  reverse  direction.    Thus  hydrogen,  iodine,  and 
hydrogen  iodide,  in  the  gaseous  state,  undergo  a  (resultant)  reaction 
hi  one  or  other  of  the  two  directions  expressed  by  the  following 
equations,  depending  on  the  concentrations  of  the  respective  sub- 
stances: 

H2+I2  =  2HI;  and  2HI=H2+I2. 

In  the  case  of  such  reactions  taking  place  simultaneously,  the  rate 
of  each  is  determined  by  its  own  specific  reaction-rate  and  by  the 
concentrations  of  the  substances  involved  in  it,  just  as  if  the  other 
reaction  were  not  taking  place.  In  accordance  with  this  law  of  in- 
dependence of  reaction-rates  the  differential  equations  expressing  the 
rates  can  be  formulated. 

Formulation  of  the  Rates  of  Simultaneous  Reactions.  — 
*Prob.  15.  —  Calculate  the  specific  reaction-rates  ki  and  k2  of  the  two 
reactions  considered  in  case  (i)  above,  supposing  that  at  some  definite 
temperature  and  with  a  definite  iodine  concentration  it  were  found 
that  in  a  CS2  solution  initially  0.5  f.  in  C6H5C1  and  0.5  f.  in  C12  15%  of 
the  C6H6C1  is  converted  into  ortho  C6H4C12  and  25%  into  para  C<jH4CW  £& 
in  30  minutes.     Ans.  £1=0.0167;  ^=0.0278. 

*Prob.  16.  —  Noting  that  in  case  (2)  the  intermediate  compound 
CH3NaC2O4  is  constantly  produced  by  the  first  reaction  and  destroyed 
by  the  second  reaction,  formulate  a  differential  equation  for  the  result- 
ant rate  dcA/dt  at  which  this  compound  is  formed,  in  terms  of  the 
initial  molalities  c0  and  2C0  of  the  (CH3)2C2O4  and  NaOH,  respectively 
and  of  the  molalities  CA  and  CB  of  (CH3)NaC2O4  and  Na2C2O4,  respec- 
tively, that  have  resulted  at  any  time  /. 


138  THE  RATE  OF  CHEMICAL  CHANGES 

Prob.  17.  —  a.  Formulate  an  expression  for  the  (resultant)  rate  at 
which  HI  is  produced  in  a  gaseous  mixture  by  the  equation  given  in 
case  (3)  in  terms  of  the  molal  concentrations  CH,  Ci,  Cm,  of  the  sub- 
stances at  any  time  t.  b.  Noting  that  there  will  be  equilibrium  when  the 
rate  of  formation  of  HI  by  the  direct  reaction  becomes  equal  to  the 
rate  of  its  decomposition  by  the  reverse  reaction,  derive  an  expression 
for  the  equilibrium  conditions  of  this  reaction. 

In  general,  for  any  reaction  represented  by  the  chemical  equation: 


the  expression  derived  (in  the  way  illustrated  by  Prob.  17)  from  the 
general  law  of  reaction-rate  (Art.  85)  for  the  equilibrium  conditions 
at  any  definite  temperature  is 

cK'cJ  .  .      ki     ^ 
—  T—  =  -r-  =  K,  a  constant. 


The  constancy  of  this  concentration  product  is  the  expression  of  the 
mass-action  law  of  chemical  equilibrium,  as  will  be  shown  in  Art.  98. 
This  result  is  of  importance  not  only  as  a  derivation  of  that  law, 
but  also  because  it  shows  that  the  equilibrium-constant  K  is  the  ratio 
ki/kz  of  the  specific  rates  of  the  two  opposing  reactions.  In  other 
words,  equilibrium  does  not  signify  a  condition  in  which  no  change 
is  taking  place,  but  one  in  which  the  two  opposing  reactions  are  taking 
place  at  equal  rates. 


THE  EFFECT  OF  CONCENTRATION  AND  SURFACE     139 

H.    THE  EFFECT  OF  CONCENTRATION  AND  SURFACE   ON  REACTIONS 
BETWEEN  SOLUTIONS   AND   SOLID   SUBSTANCES 

88.  Rate  of  Reaction  between  Solutes  and  Solid  Substances.  — 
When  a  solute  is  reacting  with  a  solid  substance  the  quantity  of  the 
solid  acted  upon  per  unit  of  time  is  proportional  to  the  surface  of  the 
solid  and  to  the  concentration  of  the  solute  that  reacts  with  it.    Thus, 
when  a  dilute  solution  of  acetic  acid  in  contact  with  a  compact  mass  of 
magnesium  oxide  is  uniformly  stirred,  the  quantity  of  the  solid  dis- 
solved is  proportional  to  the  surface  of  the  mass  and  to  the  concen- 
tration of  the  acetic  acid. 

In  reactions  with  solid  substances,  it  is  to  be  borne  in  mind  that, 
owing  to  corrosion,  the  effective  surface  is  constantly  changing,  and 
that  the  concentration  of  the  solution  in  contact  with  the  solid  is  the 
same  as  that  of  the  whole  solution  only  when  there  is  adequate  stirring. 

89.  Solid  Substances  Dissolving  in  their  Own  Solutions.  —  When 
a  solid  substance  is  dissolving  at  a  definite  temperature  in  its  own 
partially  saturated  solution,  the  rate  dc/dt  at  which  the  concentration 
of  the  solution  increases  is  proportional  to  the  difference  between  the 
saturation-concentration  5  and  the  actual  concentration  c,  and  to  the 
area  a  of  the  surface  of  the  solid,  and  it  is  inversely  proportional  to  the 
volume  v  of  the  solution;  that  is, 

dc    ,a, 


Prob.  18.  —  Rate  of  Saturation  of  a  Solution.  —  A  definite  quantity 
of  a  solid  substance  is  shaken  with  a  definite  volume  of  water  at  25°. 

a.  Calculate  the  time  of  shaking  required  to  produce  60,  90,  and  98 
percent  saturation,  assuming  that  the  solution  becomes  30%  saturated 
in  one  minute,  and  that  the  surface  of  the  solid  remains  constant. 

b.  With  the  aid  of  these  values  make  a  diagram  showing  how  the  per- 
centage saturation  increases  with  the  time  of  shaking,     c.    Find  from 
the  diagram  the  time  required  to  produce  95%  saturation,  and  the 
time  required  to  increase  the  saturation  from  95%  to  98%. 


140  THE  RATE  OF  CHEMICAL  CHANGES 

\ 

HI.    THE  EFFECT   OF  CATALYSTS 

90.  Catalysis.  —  A  reaction  is  often  greatly  accelerated  by  the 
presence  of  a  substance  which  is  not  itself  consumed  by  the  reaction. 
This  phenomenon  is  called  catalysis,  and  the  substance  producing  it  is 
called  the  catalyst. 

Although  few  general  principles  relating  to  catalysis  have  been 
established,  its  great  practical  importance  makes  it  desirable  to  con- 
sider the  common  types  of  catalysts  and  the  ways  in  which  they  act. 

..91:  Carriers.  —  Carriers  constitute  one  of  the  most  common  and 
best  understood  types  of  catalysts.  The  mechanism  of  their  action 
is  as  follows:  The  catalyst  produces  with  one  of  the  substances  an 
intermediate  compound  which  reacts  with  the  second  substance  in  such 
a  way  as  to  regenerate  the  catalyst,  the  reaction  of  the  second  sub- 
stance with  the  intermediate  compound  taking  place  more  rapidly 
than  that  with  the  first  substance.  In  this  way  a  reaction  which  does 
not  take  place  directly  at  an  appreciable  rate  may  be  made  to  take 
place  in  stages  at  a  rapid  rate.  The  chamber  process  of  making  sul- 
furic  acid  and  the  technical  method  of  making  ether  are  familiar 
examples  of  this  type,  the  fundamental  reactions  being: 

a.  O2  +  2NO  =  2NO2;  and  SO2  +  NO2+H2O=H2SO4+NO. 

b.  C2H5OH+H2SO4  =  C2H5HSO4+H2O,  and 
C2H5OH  +  C2H5HSO4  =  (C2H5)2O + H2SO4. 

The  mechanism  of  carrier  catalysis  is  well  illustrated  by  the  de- 
composition of  hydrogen  peroxide,  which  in  aqueous  solution  takes 
place  in  accordance  with  the  equation: 

2H202  =  2H20+02.  (i) 

This  decomposition  is  very  slow  when  the  hydrogen  peroxide  is  present 
alone,  but  becomes  rapid  in  the  presence  of  a  moderate  concentration 
of  iodide-ion.  The  principles  (considered  in  Probs.  19  and  20)  deter- 
mining the  rate  of  the  reaction  seem  to  show  that  this  catalytic  effect 
is  due  to  the  occurrence  of  the  reaction  in  the  two  stages  represented 
by  the  equations: 

H202+I-=H20+IO-.  (2) 

H202+IO-=  H20+02+I-  (3) 

In  the  presence  of  a  considerable  concentration  of  hydrogen-ion,  as  well 
as  of  iodide-ion,  the  hydrogen  peroxide  does  not  decompose  to  an  im- 
portant extent  into  water  and  oxygen  in  accordance  with  equation  (i), 


THE  EFFECT  OF  CATALYSTS  141 

but  it  yields  water  and  iodine  in  accordance  with  the  following 
equation: 

H202+2l-+2  H+  =  2H20+I2  (4) 

The  rate  of  this  reaction  is  found,  however,  to  be  proportional  to  the 
iodide-ion  concentration  (not  to  its  square)  and  to  be  dependent  in 
only  a  secondary  degree  on  the  concentration  of  the  hydrogen-ion. 
This  can  be  accounted  for  by  considering  that  reaction  (4)  takes  place 
in  two  stages,  namely,  that  represented  by  equation  (2)  above,  and 
that  represented  by  the  following  equation;  this  last  reaction  occurring 
almost  instantaneously  and  completely  when  the  hydrogen-ion  con- 
centration is  large: 

IO-+I-+2H+=H20+I2  (5) 

The  Mechanism  of  Carrier  Catalysis.  — 

Prob.  IQ.  —  State  and  formulate  the  principles  that  would  express  the 
rate  of  the  decomposition  of  hydrogen  peroxide  in  neutral  solution  in 
the  presence  of  iodide-ion,  assuming  that  it  takes  place,  a,  directly 
according  to  reaction  (i);  b,  according  to  reactions  (2)  and  (3),  the 
latter  taking  place  instantaneously;  c,  according  to  reactions  (2)  and 
(3),  the  IO~  produced  by  (2)  soon  attaining  such  a  concentration  that 
reaction  (3)  thereafter  takes  place  at  the  same  rate  as  reaction  (2). 
(Case  c  seems  to  represent  what  actually  occurs.) 

Prob:  20.  —  a.  Show  by  reference  to  the  equilibrium  conditions  of 
reaction  (5)  why  reaction  (3),  and  therefore  reaction  (i),  does  not  take 
place  to  any  important  extent  in  the  presence  of  a  considerable  con- 
centration of  hydrogen-ion;  also  why  reaction  (5),  and  therefore  re- 
action (4),  does  not  take  place  to  an  important  extent  in  a  nearly 
neutral  solution,  b.  Show  what  relative  quantities  of  hydrogen 
peroxide  would,  if  these  explanations  are  correct,  be  decomposed  per 
minute  in  a  solution  o.i  formal  in  H2O2  and  o.i  formal  in  KI  in  the 
two  cases  that  the  solution  is  neutral  and  that  it  is  moderately  acid 
with  HC1. 

92.  Contact  Agents.  —  Reactions  between  gases  or  solutes  are  often 
greatly  accelerated  by  placing  the  reacting  mixture  hi  contact  with 
a  suitable  solid  substance  which  offers  a  large  surface.  The  heavier 
metals  are  especially  likely  to  be  effective;  but  many  other  substances 
have  specific  effects  on  definite  reactions.  The  platinum  contact- 
process  of  making  sulfur  trioxide  from  sulfur  dioxide  and  oxygen  and  the 
Deacon  process  of  making  chlorine  by  passing  hydrogen  chloride  and 
oxygen  over  a  porous  mass  impregnated  with  copper  chloride  are 
examples  of  contact  catalysis.  As  already  stated  in  Art.  86,  gas 
reactions  are  often  catalyzed  by  solid  surfaces  to  such  an  extent  that 


142  THE   RATE  OF  CHEMICAL  CHANGES 

the  chemical  change  takes  place  appreciably  only  in  the  surface  layer 
on  the  walls  of  the  containing  vessel  or  on  other  solid  material  in 
contact  with  the  gas,  and  the  rate  of  such  reactions  does  not  conform 
to  the  mass  action  law. 

The  mechanism  of  contact  actions  is  little  understood.  In  most 
cases,  the  contact  action  is  probably  due  to  an  adsorption  of  the  re- 
acting substances  (that  is,  to  a  deposition  of  them  on  the  surface  of 
the  solid)  and  to  the  fact  that  in  the  surface-layer  the  reaction-rate 
is  greatly  increased.  Thus,  finely  divided  platinum  placed  in  contact 
with  illuminating  gas  and  air  adsorbs  the  gases,  and  these  then  react 
so  rapidly  as  to  cause  the  gas  to  take  fire. 

93.  Hydrogen-Ion  and  Hydroxide-Ion  as  Catalysts.  —  In  aqueous 
solutions  many  reactions  are  accelerated  by  hydrogen-ion.  This  is 
probably  true  of  all  reactions  in  which  water  is  directly  involved, 
such  as  the  hydrolysis  of  cane-sugar  or  of  esters.  It  is  also  true  of 
certain  reactions  of  oxidation  and  reduction. 

The  specific  rate  of  such  hydrolytic  reactions  in  dilute  solutions  is 
found  to  be  proportional  to  the  concentration  of  the  hydrogen-ion; 
for  example,  the  specific  reaction-rate  of  the  cane-sugar  hydrolysis 
at  48°  has  been  found  to  be  9.95  times  as  great  in  o.oi  normal  as  it  is  in 
o.ooi  normal  HC1  solution.  At  higher  hydrogen-ion  concentrations 
or  in  the  presence  of  neutral  salts  considerable  deviations  from  pro- 
portionality exist;  thus,  the  rate  of  the  cane-sugar  hydrolysis  is  10.5 
times  as  great  in  o.i  normal  as  in  o.oi  normal  HC1  solution. 

This  principle  can  be  employed  (as  shown  by  Prob.  22)  for  deter- 
mining the  hydrogen-ion  concentration  in  solutions;  for  no  other 
ion  (except  hydroxide-ion  in  certain  cases)  exerts  a  catalytic  effect  on 
hydrolytic  reactions. 

Reactions  in  which  water  takes  part  are  often  accelerated  also  by 
hydroxide-ion;  thus,  milk-sugar,  C^H^On,  dissolved  in  water  becomes 
hydrated  (with  formation  of  Ci2H22Oii.H2O)  at  a  rate  which  is  greatly 
increased  by  hydroxide-ion. 

Prob.  21.  —  Possible  Explanation  of  the  Catalytic  Effect  of  Hydrogen- 
Ion.  —  Show  by  writing  chemical  equations  how  the  catalytic  effect 
of  hydrogen-ion  on  hydrolytic  reactions  may  be  interpreted  as  a  carrier 
action,  assuming  that  the  hydrogen-ion  is  hydrated. 

Prob.  22.  —  Determination  of  Hydrogen-Ion  Concentration  through  its 
Catalytic  Effect.  —  Diazoacetic  ester  decomposes  in  aqueous  solution  ac- 
cording to  the  equation  CHN2.CO2C2H6+H2O  =  CH2OH.CO2C2H6+N2, 
and  the  reaction  is  catalyzed  by  hydrogen-ion.  At  25°  in  a  solution  o.i 


THE  EFFECT  OF  CATALYSTS  143 

formal  in  acetic  acid,  whose  ionization  is  1.34%,  it  is  found  by  measuring 
the  nitrogen  evolved  that  37.5%  of  the  ester  is  decomposed  in  10 
minutes.  Assuming  that  it  takes  67  minutes  to  decompose  the  same 
percentage  of  the  ester  in  a  solution  o.i  formal  in  sodium  hydrogen 
tartrate,  calculate  the  molality  of  hydrogen-ion  in  that  solution. 

94.  Enzymes.  —  Certain  complex  organic  substances,  called  en- 
zymes, which  are  produced  by  animal  and  plant  organisms,  have  an 
extraordinary  power  of  catalyzing  certain  organic  'reactions.    The 
effect  is  highly  specific,  a  particular  enzyme  being  required  for  a 
particular  reaction      Thus  invertase,  an  enzyme  produced  by  yeast, 
causes  the  conversion  of  cane-sugar  into  glucose  and  fructose;  and 
xymase,  another  yeast  enzyme,  causes  the  conversion  of  glucose,  but 
not  of  the  analogous  compound  fructose,  into  ethyl  alcohol  and  carbon 

dioxide. 

«•  -»      «* '  t 

95.  Water  as  a  Catalyst.  —  The  presence  of  water  hi  at  least  minute 

quantity  is  essential  to  the  occurrence  of  almost  all  reactions.  This 
is  shown  by  experiments  upon  some  of  the  most  energetic  chemical 
changes,  which  are  found  not  to  take  place  when  the  separate  sub- 
stances are  very  thoroughly  dried  before  they  are  brought  together. 
Thus,  this  is  true  of  the  combination  of  sodium  with  chlorine,  the 
union  of  ammonia  and  hydrogen  chloride  gases,  the  combustion  of 
carbon  monoxide  with  oxygen,  and  the  union  of  lime  and  sulfur 
trioxide. 


144  THE  RATE  OF  CHEMICAL  CHANGES 

IV.    THE  EFFECT  OF   TEMPERATURE 

96  Principles  Relating  to  the  Effect  of  Temperature.  —  Measure- 
ments of  the  change  with  the  temperature  of  the  rates  of  reactions 
taking  place  between  solutes  or  between  gases  have  shown  that  as  a 
rule  equal  small  increments  of  temperature  cause  an  approximately 
equal  multiplication  of  the  specific  rate  of  any  definite  reaction.  Thus, 
if  the  specific  rate  of  a  reaction  is  increased  2.5  fold  by  raising  the 
temperature  from  o  to  10°,  it  will  again  be  increased  approximately 
2.5  fold  by  raising  the  temperature  from  10  to  20°.  This  principle 
and  the  order  of  magnitude  of  the  deviations  from  it  are  illustrated 
by  the  following  data  for  the  third-order  reaction  between  ferrous 
chloride,  potassium  chlorate,  and  hydrochloric  acid.  The  values 
ki/kz  are  the  ratios  of  the  specific  reaction-rates  at  10°  intervals. 

IO  tO  0°      20  tO  10°      30  tO  20°      40  tO  30°      50  tO  40° 
&1/&2  2.8  2.7  2.5  2.4  2.2 

A  more  general  and  more  exact  expression  of  the  effect  of  tempera- 
ture is  that  given  by  the  following  equations,  in  which  A  is  a  constant 
for  any  definite  reaction: 

7 ,      T       .  dT         ,     kz      *  Tz —  TI 
rflog^-jorlog^^. 

It  is  evident  that,  so  long  as  the  variation  of  T\  Tz  is  not  large,  the 
approximate  principle  is  a  consequence  of  this  more  general  law. 

Prob.  23.  —  The  Effect  of  Temperature.  —  a.  In  a  solution  o.oi  molal 
in  sodium  hydroxide  and  o.oi  molal  in  ethyl  acetate,  39%  of  the  ethyl 
acetate  is  decomposed  in  10  minutes  at  25°,  and  55%  at  35°.  Deter- 
mine by  the  approximate  principle  how  long  it  would  take  to  decompose 
55%  at  15°.  b.  Calculate  by  the  logarithmic  equation  how  long  it 
would  take  to  decompose  50%  of  the  ethyl  acetate  at  20°. 

It  has  also  been  found  that  in  the  case  of  different  reactions  equal 
small  increments  of  temperature  cause  not  far  from  the  same  multipli- 
cation of  their  specific  rates.  Thus,  a  10°  rise  of  temperature  multi- 
plies the  specific  rate  of  the  reaction  between  sodium  hydroxide  and 
ethyl  acetate  1.9  fold,  of  that  between  cane-sugar  and  water  in  the 
presence  of  acid  3.6  fold,  and  of  that  between  ferrous  sulfate, 
potassium  chlorate,  and  sulfuric  acid  2.4  fold. 

Prob.  24.  —  Uniform  Effect  of  Temperature  on  Different  Reactions. — 
At  100°  a  certain  reaction  takes  place  to  an  extent  of  25%  in  one  hour. 
Estimate  roughly  how  long  it  would  take  for  the  reaction  to  proceed  to 
the  same  extent  at  20°. 


CHAPTER  VII 

THE  EQUILIBRIUM   OF   CHEMICAL   CHANGES  AT 
CONSTANT  TEMPERATURE 


I.    THE   LAW  OF  MASS-ACTION 

97.  The  Equilibrium  of  Chemical  Reactions.  —  When  two  or  more 
chemical  substances  capable  of  reacting  with  one  another  are  brought 
together,  it  is  always  true  after  a  sufficiently  long  time  (which  may 
vary  from  a  fraction  of  a  second  to  thousands  of  years)  that  the  chemi- 
cal reaction  which  has  been  taking  place  between  them  practically 
ceases  —  in  other  words,  that  a  condition  is  reached  where  no  further 
change  takes  place.  The  reaction  is  then  said  to  be  in  equilibrium. 

Many  reactions  (for  example,  the  reaction  NH4OH+HBO2  = 
NH4BO2+H2O  in  aqueous  solution)  take  place  so  incompletely  that 
at  equilibrium  the  substances  on  both  sides  of  the  equation  are  present 
in  measurable  proportions.  But  with  many  other  reactions  the 
equilibrium-conditions  are  such  that  the  change  seems  to  take  place 
completely  when  the  substances  on  one  side  of  the  chemical  equation 
expressing  it  are  brought  together,  and  not  to  take  place  at  all  when 
the  substances  on  the  other  side  are  brought  together;  thus,  this  is 
true  of  the  reaction  NaOH+HCl  =  NaCl-|-H2O  in  aqueous  solution. 
In  all  such  cases,  however,  the  gaseous  or  dissolved  substances  on 
both  sides  of  the  equation  are  really  present  at  some  concentration, 
though  this  may  be  so  small  in  the  case  of  some  of  the  substances  that 
it  cannot  be  directly  measured. 

Reactions  (like  that  in  a  mixture  of  hydrogen  and  oxygen  at  25°) 
which  are  taking  place  so  slowly  that  no  appreciable  change  can  be 
detected  within  a  reasonable  time  are  not  to  be  confounded  with  those 
which  are  in  a  state  of  equilibrium.  Whether  equilibrium-conditions 
have  been  attained  can  be  determined  by  causing  the  reaction  to  take 
place  in  the  two  opposite  directions  and  comparing  the  concentrations 
of  the  substances  in  the  two  cases  after  equilibrium  seems  to  have 
been  reached. 

The  equilibrium  conditions  of  chemical  reactions  vary  with  the 
temperature,  as  considered  in  Chapter  XII.  In  this  chapter  the 
principles  will  be  discussed  which  determine  equilibrium  when  the 
temperature  has  any  constant  value. 

US 


146  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

98.  The  Mass-Action  Law  of  Chemical  Equilibrium.  —  The  effect 
of  concentration  in  determining  chemical  equilibrium  is  expressed 
by  one  of  the  most  fundamental  laws  of  chemistry.  This  law,  which  is 
commonly  known  as  the  mass-action  law,  states  that,  whatever  be  the 
initial  concentrations  of  the  gaseous  or  dissolved  substances  A,  B,  ... 
E,  F, . . .  involved  in  any  definite  chemical  reaction,  such  as  may  be 
represented  in  general  by  the  equation,  0A+6B.  . .  =  eE+/F. . .,  the 
reaction  always  takes  place  in  such  a  direction  and  to  such  an  extent 
that,  when  equilibrium  is  reached  at  any  definite  temperature,  the 
conditions  are  satisfied  which  are  expressed  by  the  equation: 

C*C  .'  "  =  K,  a  constant. 
cAac»b... 

In  this  expression  CE,  cp,  .  .  .  CA,  cz,  . . .  denote  either  the  molal  concen- 
trations or  molalities  of  the  substances  E,  F,  .  .  .  A,  B,.  .  .  in  the  equi- 
librium mixture,  and  e,  /,.  .  .  a,  br  .  .  denote  the  number  of  mols  of  them 
that  are  involved  in  the  reaction  expressed  by  the  chemical  equation.  A 
simpler  notation  commonly  employed  is  to  represent  the  molalities 
of  solutes  at  equilibrium  by  their  formulas  enclosed  within  paren- 
theses; thus,  by  writing  (E)  in  place  of  CE,  etc. 

The  quantity  K,  which  is  a  constant  characteristic  of  the  reaction, 
is  called  its  equilibrium-constant.  Its  value  is,  of  course,  constant 
with  respect  to  variations  of  the  initial  and  equilibrium  concentra- 
tions, but  it  varies  with  the  temperature.  In  evaluating  it,  molal 
concentrations  will  be  used  in  the  case  of  gases,  and  molalities  (de- 
fined as  in  Art.  35)  in  the  case  of  solutions.  It  is  customary  to  place 
in  the  numerator  the  concentrations  of  the  substances  occurring  on 
the  right-hand  side  of  the  chemical  equation  written  in  some  specified 
way.  In  which  direction  the  chemical  equation  is  written  is,  of 
course,  arbitrary  in  the  case  of  reactions  at  equilibrium.  With  many 
types  of  reactions,  however,  it  is  more  natural  to  write  the  equation 
in  one  of  the  two  directions;  and  in  such  cases  the  usage  in  evaluating 
the  equilibrium-constant  has  become  fairly  definite.  This  usage  will 
be  illustrated  by  the  examples  given  in  the  following  articles. 

The  mass-action  law  of  equilibrium  is  a  consequence  of  the  mass- 
action  law  of  reaction-rate,  as  shown  in  Art.  87.  It  can  also  be  derived 
with  the  aid  of  the  laws  of  thermodynamics  from  the  physical  laws  of 
perfect  gases  and  of  perfect  solutions,  as  shown  in  Arts.  140  and  141. 
It  has  also  been  established  by  experimental  investigations  on  the 
equilibrium  of  a  large  number  of  different  reactions. 


THE  LAW  OF  MASS-ACTION  147 

The  mass-action  law  is  exact,  as  is  shown  by  its  derivation,  only 
in  the  case  of  reactions  between  perfect  gases  or  perfect  solutes;  but 
it  holds  true  approximately  when  applied  to  gases  at  moderate  pres- 
sures, to  unionized  solutes  at  moderate  concentrations,  and  to  ions 
at  small  concentrations.  The  deviations  from  it  in  such  cases  may  be 
expected  to  be  of  the  same  order  of  magnitude  as  those  from  the 
physical  laws  of  perfect  gases  or  perfect  solutions. 

Although  the  mass-action  law  of  chemical  equilibrium  has  been 
derived  from  the  mass-action  law  of  reaction-rate,  an  important 
difference  between  the  two  laws  is  to  be  noted.  Namely,  it  can  be 
shown  that,  although  the  expression  for  the  rate  of  a  reaction  depends 
on  its  mechanism,  the  same  expression  is  obtained  for  its  equilibrium, 
whatever  be  the  process  by  which  that  equilibrium  is  considered  to  be 
attained.  In  using  the  equilibrium  expression  it  is  necessary  to  know 
only  the  concentrations  of  the  substances  actually  occurring  in  that 
expression. 


148  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

II.    THE  MASS-ACTION  LAW  FOR  REACTIONS   BETWEEN   GASES 

99.  The  Mass-Action  Law  in  Terms  of  Partial  Pressures.  —  In 
applications  of  the  mass-action  law  to  gases,  it  is  usual  to  substitute 
for  the  concentrations  of  the  substances  in  the  equilibrium-mixture 
their  partial  pressures.  This  is  admissible  since  at  any  definite 
temperature  the  two  quantities  are  proportional  to  one  another  in 
the  case  of  perfect  gases,  in  virtue  of  the  relation  p=cRT. 

The  general  mass-action  expression  in  terms  of  partial  pressures 
evidently  is 


where  Kp  is  a  constant,  called  the  equilibrium-constant  in  terms  of 
pressure,  which  has  in  general  a  different  numerical  value  from  the 
constant  K  occurring  in  the  corresponding  concentration-expression. 
In  evaluating  it  the  partial  pressures  are  commonly  expressed  in 
atmospheres. 

Prob.  i.  —  Relation  between  the  Equilibrium  Expressions  in  Terms  of 
Concentrations  and  Pressures.  —  Derive  the  general  mass-action  expres- 
sion in  terms  of  pressure  from  that  in  terms  of  concentration,  and  show 
the  relation  between  the  two  equilibrium-constants. 

100.  Gaseous  Dissociation.  —  A  chemical  change  which  consists  in 
the  splitting  of  a  substance  with  complex  molecules  into  one  or  more 
substances  with  simpler  molecules  is  called  dissociation.  Thus, 
the  reactions  I2  =  2l,  NH4C1  =  NH3+HC1,  and  2CO2  =  2CO+O2, 
are  examples  of  dissociation.  The  fractional  extent  to  which  the 
dissociating  substance  has  been  decomposed  is  called  its  degree  of 
dissociation,  or  simply  its  dissociation  (7).  The  equilibrium-constant 
of  such  a  reaction  is  commonly  called  the  dissociation-constant  of  the 
dissociating  substance. 

It  is  characteristic  of  such  reactions  that  the  number  of  molecules 
increases  when  the  dissociation  increases.  Since  the  pressure-volume 
product  of  gases  increases  correspondingly,  the  degree  of  dissociation 
can  always  be  determined  by  measuring  the  volume  or  density  of  the 
gas  at  a  known  temperature  and  pressure,  and  /  comparing  it  with 
that  calculated  for  the  undissociated  or  completely  dissociated  sub- 
stance, as  is  illustrated  by  Probs.  10  and  n  of  Art.  12,  and  by  Prob. 
3  below. 

Another  common  method  of  determining  the  composition  of  the 
equilibrium  mixture  is  to  cool  it  suddenly  to  a  lower  temperature  at 


REACTIONS  BETWEEN  GASES  149 

which  the  reaction-rate  is  so  small  that  the  original  equilibrium  is  not 
displaced,  and  then  to  analyze  the  mixture.  This  method  presupposes 
that  there  is  no  change  hi  the  composition  during  the  short  period  of 
cooling.  This  condition  may  be  practically  realized  in  cases  where 
the  rate  at  which  the  equilibrium  is  established  is  comparatively  slow 
even  at  the  higher  temperature,  or  in  cases  where  the  reaction  takes 
place  only  in  contact  with  a  catalyst.  In  the  latter  cases  the  equi- 
librium-mixture can  be  separated  from  the  catalyst,  and  subsequently 
cooled  without  the  danger  of  any  change  taking  place. 

Applications  of  the  Mass- Action  Law  to  Gaseous  Dissociation.  — 
Prob.  2.  —  a.   Formulate  the  mass-action  expression  for  the  dis 
sociation  of  sulfur  trioxide  into  sulfur  dioxide  and  oxygen,     b.  At  630° 
and  i  atm.  the  sulfur  trioxide  is  just  one  third  dissociated.     Calculate 
the  dissociation-constant  of  sulfur  trioxide. 

Prob.  3.  —  Show  how  the  dissociation  (7)  of  sulfur  trioxide  could  be 
calculated  from  measurements  of  the  density  (d)  of  the  equilibrium 
mixture  at  the  temperature  (T)  and  pressure  (p). 

Prob.  4.  —  A  mixture  consisting  of  i  mol  of  SO2  and  i  mol  of  O2  is 
passed  at  630°  and  i  atm.  through  a  tube  containing  finely  divided 
platinum  so  slowly  that  equilibrium  is  attained,  and  the  issuing  gas  is 
cooled  and  analyzed  by  absorbing  the  sulfur  dioxide  and  trioxide  by 
potassium  hydroxide  and  measuring  the  residual  oxygen  gas.  At  o° 
and  i  atm.  the  volume  of  this  residual  gas  was  found  to  be  13,780  ccm., 
corresponding  to  0.615  mol.  a.  Calculate  the  dissociation-constant  of 
sulfur  trioxide.  b.  Calculate  the  ratio  of  the  mols  of  sulfur  trioxide 
to  the  mols  of  sulfur  dioxide  in  an  equilibrium  mixture  at  630°  in  which 
the  partial  pressure  of  oxygen  is  0.25  atm. 

Prob.  5.  —  At  a  certain  temperature  a  definite  quantity  of  phosphorus 
pentachloride  gas  has  at  i  atm.  a  volume  of  i  liter,  and  under  these 
conditions  it  is  about  50%  dissociated  into  PC13  and  C12.  Show  by 
reference  to  the  mass-action  expression  whether  the  dissociation  will 
be  increased  or  decreased:  a,  when  the  pressure  on  the  gas  is  reduced 
till  the  volume  becomes  2  1.;  6,  when  nitrogen  is  mixed  with  the  gas 
till  the  volume  becomes  2  1.,  the  pressure  being  still  i  atm.;  c,  when 
nitrogen  is  mixed  with  the  gas  till  the  pressure  becomes  2  atm.,  the 
volume  being  still  i  1.;  rf,  when  chlorine  is  mixed  with  the  gas  till  the 
pressure  becomes  2  atm.,  the  volume  being  still  i  1.;  e,  when  chlorine  is 
mixed  with  the  gas  till  the  volume  becomes  2  1.,  the  pressure  being  still 
i  atm.  —  In  answering  these  questions  consider  whether  the  first  effect 
of  the  change  in  conditions  (assuming  that  no  reaction  takes  place)  is 
to  increase  or  decrease  the  value  of  the  ratio  pc\t  ppci,/ppc\s,  and  in 
which  direction  the  reaction  must  take  place  in  order  to  restore  the 
equilibrium-value  of  this  ratio. 

Prob.  6.  —  a.  Derive  for  the  dissociation  of  water-vapor  into  hydrogen 
and  oxygen  a  mass-action  expression  which  will  show  how  the  dissocia- 


150  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

tion  7  varies  with  the  total  pressure  p.  b.  At  2000°  the  dissociation 
is  2.0%  when  the  total  pressure  is  i  atm.  Calculate  the  dissociation 
when  the  total  pressure  is  0.33  atm.  (as  it  is  approximately  in  the 
gaseous  mixture  produced  by  burning  hydrogen  with  the  minimum 
amount  of  air).  Solve  the  equation  approximately,  neglecting  the 
(small)  value  of  the  dissociation  where  this  is  justifiable.  Ans.  6,  2.9%. 

101.  Metathetical  Gas  Reactions.  —  Examples  of  metathetical 
gas  reactions  whose  equilibrium  has  been  investigated  are: 

CO2+H2  =  CO+H2O,  and  4HC1+02  =  2C12+2H2O. 

An  important  principle  in  regard  to  them,  illustrated  by  Prob.  8 
below,  is  that  the  equilibrium-constant  of  any  metathetical  gas  re- 
action can  be  calculated  from  the  dissociation-constants  of  the  com- 
pounds involved  in  it. 

Another  important  principle  relates  to  the  effect  of  pressure.  It 
states  that  increase  of  pressure  causes  the  equilibrium  of  any  gaseous 
reaction  to  be  displaced  in  that  direction  in  which  the  number  of 
molecules,  and  therefore  the  volume  of  the  gas,  decreases.  This  prin- 
ciple has  already  been  illustrated  by  the  fact  that  dissociation  is 
decreased  by  increase  of  pressure.  It  is  demonstrated  in  Prob.  9. 

Prob.  7.  —  Combustion  of  "Water-Gas"  with  Insufficient  Oxygen.  — 
The  equilibrium-constant  of  the  reaction  CO2+H2  =  CO+H2O  at 
1120°  is  2.0.  Calculate  the  ratio  of  the  mols  of  CO2  to  the  mols  of  H2O 
that  are  formed  when  a  "water-gas"  consisting  of  i  mol  of  CO  and  i 
mol  of  H2  is  burnt  at  1120°  with  |  mol  of  O2.  Assume  that  equilibrium 
is  attained  and  that  the  quantity  of  the  oxygen  which  remains  un- 
combined  is  negligible. 

Prob.  8.  —  Metathetical  Equilibrium-Constants  in  Relation  to  Dissocia- 
tion-Constants. —  The  equilibrium-constant  of  the  reaction  2CO2  = 
2CO+O2  at  1120°  is  i.4Xio-12.  a.  Calculate  the  partial  pressure  of 
the  oxygen  in  the  equilibrium  mixture  of  Prob.  7.  b.  Calculate  the 
dissociation-constant  of  water- vapor  at  1120°.  c.  Show  what  relation 
exists  between  the  dissociation-constants  Kw  and  #002  of  water-vapor 
and  of  carbon  dioxide  and  the  equilibrium-constant  K  of  the  reaction 
CO2+H2  =  CO+H2O.  Ans.  6,^.5  Xio-1'. 

Prob.  g.  —  Effect  of  Pressure  on  Gaseous  Equilibrium.  — •  Prove  that 
the  equilibrium  of  any  chemical  reaction,  aA+6B=eE-f/F,  must  be 
displaced  in  the  direction  in  which  the  number  of  molecules  decreases 
when  the  total  pressure  p  of  the  equilibrium  mixture  (in  which  the 
substances  are  present  at  mol-fractions  XA,  XB,  #E,  and  X?)  is  increased. 

Prob.  10.  —  Production  of  Hydrogen  from  "Water-Gas."  —  A  quantity 
of  "water-gas"  consisting  of  i  mol  of  CO  and  i  mol  of  H2  is  mixed  with 
5  mols  of  steam,  and  this  mixture  is  passed  slowly  at  i  atm.  and  500° 
over  a  suitable  catalyst,  whereby  the  reaction  CO+H2O  =  CO2+H2 


REACTIONS  BETWEEN  GASES  151 

takes  place.  The  gases  are  then  cooled  in  order  to  condense  the  steam. 
The  equilibrium-constant  of  the  reaction  at  500°  is  5.5.  a.  Calculate 
the  smallest  mol-fraction  of  CO  in  the  resulting  gas  mixture  attainable 
by  this  process,  b.  Show  whether  this  minimum  percentage  of  CO 
could  be  diminished  by  passing  the  gas  mixture  over  the  catalyst  at 
5  atm.  and  500°.  c.  Show  whether  it  could  be  diminished  by  passing 
it  over  the  catalyst  at  i  atm.  and  400°,  noting  the  values  of  the  equi- 
librium-constant already  given  at  1120°  and  500°.  d.  State  the  effect 
of  the  lower  temperature  on  the  time  required  for  reducing  the  per- 
centage of  CO  to  any  definite  value,  e.  In  practice  the  above  process 
may  yield  a  mixture  containing  4  mol-percent  of  CO,  65  mol-percent 
of  H2,  and  31  mol-percent  of  CO2.  To  free  this  mixture  from  CO2,  it  is 
passed  at  30  atm.  up  a  tower  through  which  water  at  17°  is  trickling. 
Calculate  the  minimum  volume  of  water  theoretically  required  to  remove 
99%  of  the  CO2  from  100  mols  of  the  gas  mixture.  Assume  Henry's 
law  to  hold  true,  and  take  the  solubility  of  CO2  hi  water  at  17°  and 
i  atm.  as  0.0425  molal.  /.  Calculate  the  percentage  of  the  hydrogen 
that  is  lost  by  dissolving  in  the  water,  from  the  fact  that  the  solubility 
of  hydrogen  gas  in  water  at  17°  and  i  atm.  is  0.00084  molal.  Ans. 
e,  77.7  liters. 


EQUILIBRIUM  OF  CHEMICAL  CHANGES 


/ 


III.    THE   MASS-ACTION   LAW  FOR  REACTIONS   BETWEEN   SOLUTES 

102.  lonization  of  Slightly  Ionized  Univalent  Acids  and  Bases.  — 
The  mass-action  law  has  been  found  to  be  applicable  to  the  ionization 
of  the  slightly  ionized  monobasic  acids  and  monacidic  bases.  This  is 
illustrated  by  the  following  values  at  18°  of  the  ionization,  taken 
equal  to  the  conductance- viscosity-ratio,  and  of  the  ionization- 
constant  K  for  ammonium  hydroxide,  that  is,  the  equilibrium-constant 
of  the  reaction  NH4OH  =  NH4++OH-. 

IONIZATION  OF  AMMONIUM  HYDROXIDE  AT  18° 
Formality      0.500       0.300       o.ioo       o.oio       o.ooi 
Ionization      0.0056      0.0074     0.0130      0.0405      0.123 
KXio6  15.8          16.6          17.1          17.1        17.1 

The  ionization-constant  varies  greatly  with  the  composition  and 
structure  of  the  acid  or  base,  as  will  be  seen  from  the  f ollowing  values. 

lONIZATION-CONSTANTS  OF  ACIDS   AND  BASES   AT   25° 


Acid 

I0"£ 

Acid                    io«K 

Acid 

Values  of  io6K 

HCN 

0.0005 

HC02H             210 

C6H5CO2H 

60 

60 

60 

HBO2 

0.0017 

CH3CO2H           18 

Ortho 

Meta 

Para 

HC1O 

0.044 

C2H5C02H          13 

C6H4OHCO2H 

1020 

87 

29 

HNO2 

400. 

n-C3H7C02H       15 

C6H4C1CO2H 

1320 

155 

93 

HF 

790. 

CH2C1C02H   1550 

C6H4NO2C02H 

6l6o 

345 

396 

Base                                     icfiK 

NH4OH                      18. 

CH3NH3OH             500. 

C6H6NH3OH                0.0004 

Prob.  ii.  —  Derivation  of  lonization-Constants  from  Conductanci 
Data.  —  a.  Calculate  the  ionization-constant  of  acetic  acid  at  18°  a| 
the  concentrations  o.oi,  o.i,  and  i.o  formal, from  the  conductance  values 
given  in  Art.  74.  b.  Calculate  the  ionization-constants  at  o.i  and  i.o 
formal,  taking  into  account  the  fact  that  the  viscosity  of  the  i  formal 
solution  is  1.113  times  as  great  as  that  of  pure  water,  c.  Calculate 
the  ionization  of  the  acid  and  its  equivalent  conductance  in  o.ooi  formal 
solution. 

Prob.  12.  —  Effect  of  the  Presence  of  a  Substance  with  a  Common  Ion.  — 
a.  Show  that  the  hydrogen-ion  concentration  in  an  acetic  acid 
solution  is  decreased  by  the  addition  of  sodium  acetate  approximately 
in  the  proportion  in  which  the  concentration  of  the  acetate-ion  is  in- 
creased, b.  Calculate  the  ionization  of  acetic  acid  in  a  solution  o.i 
formal  both  in  acetic  acid  and  in  sodium  acetate  at  18°,  regarding  the 
sodium  acetate  as  completely  ionized.  —  In  this  and  other  mass-action 
problems  make  any  simplifications  that  will  not  produce  in  the  result 
an  error  greater  than  i%. 


REACTIONS  BETWEEN  SOLUTES  153 

103.  lonization  of  Largely  Ionized  Substances.  —  As  was  stated  in 
Art.  79,  the  conductance-ratio  of  salts  and  largely  ionized  acids  and 
bases  does  not  change  with  the  concentration  even  approximately 
as  the  mass-action  would  require  if  that  ratio  were  equal  to  the  ioniza- 
tion.  Now  that  the  mass-action  law  has  been  formulated,  the  degree 
of  the  divergence  can  be  shown  quantitatively.  The  following  table 
contains  the  values  (designated  7  observed)  of  the  conductance- 
viscosity  ratio  for  potassium  chloride,  the  ionization-constants  (K) 
calculated  from  them  by  the  mass-action  expression  (7  c)2/(i  — T)C, 
and  the  ionization-values  (7  calculated)  computed  conversely  from 
the  value  of  the  ionization-constant  at  o.i  formal. 

THE  CONDUCTANCE-RATIO  IN  RELATION  TO  THE  MASS-ACTION  LAW 

Formality    .      .      .  o.ooi  o.oi  0.05  o.io  0.20  0.50  i.oo 

7  observed  .      .      .  0.982  0.94  0.89  0.86  0.83  0.78  0.74 

7  calculated       .      .  0.998  0.98  0.92  0.86  0.78  0.63  0.51 

Ionization-constant  0.053  0.15  0.36  0.53  0.81  1.38  2.10 

.  This  striking  discordance  with  the  mass-action  law  and  certain 
other  anomalies,  some  of  which  were  discussed  in  Art.  79,  have  led,  as 
already  stated,  to  the  conclusion  that  the  conductance-ratio  or  the 
conductance-viscosity-ratio  is  not  even  an  approximate  measure  of 
the  mass-action  of  the  ions  or  unionized  molecules  of  largely  ionized 
substances.  It  will  be  shown  in  Art.  113  that  there  is  a  general 
method,  differing  in  principle  from  the  conductance  method,  by 
which  there  can  be  derived  the  mass-action  of  substances  (that  is, 
their  effect  in  determining  equilibria)  in  cases  where  the  mass-action  • 
is  not  proportional  to  the  concentration  as  the  mass-action  law  requires. 
The  data  needed  in  order  to  calculate  by  this  method  the  mass-action 
of  such  imperfect  solutes  are,  however,  very  incomplete;  and  the 
following  simple  hypotheses  lead  to  results  which  usually  do  not  differ 
by  more  than  a  few  percent  from  the  results  derived  with  the  aid  of 
the  exact  principle,  in  the  case  of  solutes  not  more  concentrated  than 
i  formal.  These  hypotheses  are  that  largely  ionized  substances  are 
completely  ionized  and  that  their  ions  act  as  perfect  solutes.  These 
hypotheses  are  therefore  employed  throughout  this  chapter  in  all  the 
applications  of  the  mass-action  law  to  the  different  types  of  equi- 
librium between  dissolved  substances,  and  they  are  to  be  used  in 
solving  all  the  problems.  In  Art.  113  at  the  end  of  this  chapter  a 
method  suitable  for  the  exact  treatment  of  ordinary  solutes  at  large 
concentration,  and  of  ions  at  small  concentrations,  will  be  described. 


154  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

104.  The  lonization  of  Water.  —  The  dissociation  of  water  into  H+ 
and  OH~  ions  is  extremely  small,  as  is  shown  by  the  very  small  con- 
ductance of  the  purest  water  that  has  been  prepared.    This  ionization 
is,  nevertheless,  of  great  significance  from  an  equilibrium  viewpoint, 
since  water  is  involved  in  many  ionic  reactions,  for  example,  in  the 
hydrolysis  of  salts  considered  in  Art.  105. 

The  complete  expression  for  the  equilibrium-constant  of  the  reaction 
H2O  =  H+-f-OH~  would  contain  a  term  representing  the  mass-action 
of  the  chemical  substance  H2O.  In  the  case  of  a  substance  so  con- 
centrated, it  is  not  justifiable  to  represent  its  mass-action  by  its  con- 
centration, since  the  mass-action  law  is  applicable  only  to  substances 
at  small  concentrations.  It  will,  however,  be  shown  in  Art.  113  that 
the  mass-action  of  a  substance  in  a  solution  of  any  concentration  is 
proportional  to  its  vapor-pressure  in  that  solution;  hence  the  equi- 
librium-constant for  the  ionization  of  water  is  accurately  expressed 
by  the  equation  J£  =  (H+)(OH~)/^H20-  -Now,  since  the  vapor-pres- 
sure of  water  in  a  i  molal  solution  of  any  solute  is  according  to  Raoult's 
law  only  1.8  percent  less  than  that  of  pure  water,  the  mass-action  of 
the  water  is  nearly  constant  in  aqueous  solutions  of  moderate  con- 
centration; and  the  equilibrium-constant  of  the  reaction,  commonly 
called  the  ionization-constant  Kw  of  water,  may  be  written  in  the 
simple  form:  Kw  =  (H+)  (OH~). 

The  ionization-constant  of  water  has  been  determined  by  a  number 
of  different  methods  including  that  based  on  the  conductance  of  care- 
fully purified  water,  illustrated  by  Prob.  34  of  Art.  77.  Its  value  has 
been  found  to  be  o.nXio~14ato°;  i.ooXio~14at  25°;  and48Xio~14 
at  100°. 

From  the  value  of  this  ionization-constant  there  can  be  calculated 
the  concentration  of  hydrogen-ion  in  any  dilute  aqueous  solution  of 
known  hydroxide-ion  concentration,  and  conversely. 

Prob.  ij.  —  Concentration  of  Hydrogen-Ion  in  Aqueous  Solutions.  — 
Calculate  the  concentration  of  hydrogen-ion  at  25°,  a,  in  pure  water; 
b,  in  o.i  formal  NaOH  solution. 

105.  The  Hydrolysis  of  Salts.  —  When  either  the  acid  or  base  of  a 
salt  has  a  very  small  ionization-constant,  the  salt  in  aqueous  solution 
reacts  with  the  water  to  an  appreciable  extent  with  formation  of  the  acid 
and  base.    This  phenomenon  is  called  hydrolysis;  and  the.  fraction 
of  the  salt  hydrolyzed  is  called  the  degree  of  hydrolysis,  or  simply  the 
hydrolysis  (ti).    Thus,  at  25°  in  o.oi  formal  solution  potassium  cyanide 


REACTIONS  BETWEEN  SOLUTES  155 

and  ammonium  cyanide  are  hydrolyzed  to  the  extent  of  4.4  percent 
and  51.3  percent,  respectively,  according  to  the  reactions: 

K+CN-+H2O  =K+OH-+HCN. 
NH4+CN-+H20  =NH4OH+HCN. 

As  is  done  in  the  preceding  chemical  equations,  it  is  convenient  to 
indicate  largely  ionized  substances,  whose  ionization  is  unknown  and 
as  an  approximation  may  be  assumed  to  be  complete,  by  attaching 
+  and  —  signs  to  their  ions,  and  to  indicate  slightly  ionized  substances, 
whose  ionization  conforms  to  the  mass-action  law,  by  omitting  these 
signs. 

The  hydrolysis  of  salts  can  be  experimentally  determined  by  a 
variety  of  methods.  It  may  be  derived  from  measurements  of  the 
conductance  of  the  salt,  as  in  Prob.  35  of  Art.  77,  and  in  Prob.  14 
below;  also  from  the  effect  of  the  salt  on  the  rate  of  reactions  whose 
rates  are  proportional  to  the  concentration  of  hydroxide-ion  or  hy- 
drogen-ion in  the  solution,  as  in  Prob.  12  of  Art.  85  and  in  Prob.  15 
below.  It  may  also  be  determined  by  measuring  the  distribution 
of  the  slightly  ionized  acid  or  base  of  the  salt  into  an  organic  solvent, 
as  in  Prob.  16  below. 

Prob.  14.  —  Determination  of  Hydrolysis  by  Conductance  Measure- 
ments. —  The  specific  conductance  at  100°  of  a  0.025  formal  solution  of 
NH4C2H3O2  (ammonium  acetate)  is  0.00685  reciprocal  ohms;  and  that 
of  a  second  solution  0.025  formal  in  NILQHsC^  and  0.025  formal  in 
NH4OH  is.  0.00717  reciprocal  ohms.  Calculate  the  hydrolysis  of  the 
ammonium  acetate  in  the  first  solution,  assuming  that  in  the  second 
solution  the  hydrolysis  of  the  salt  has  been  reduced  to  zero  by  the  excess 
of  base  present  and  that  the  ionization  and  conductance  of  the  base  in 
the  presence  of  its  neutral  salts  are  negligible. 

Prob.  15.  —  Determination  of  Hydrolysis  by  Reaction-Rate  Experi- 
ments. —  The  specific  reaction-rate  at  100°  of  the  sugar  hydrolysis  has 
been  found  to  be  0.0386  in  a  solution  o.ooi  formal  in  HC1,  and  to  be 
0.0946  in  a  solution  o.oi  formal  in  AlCls.  Calculate  the  fraction  of  the 
salt  that  is  hydrolyzed  (into  A1(OH)3  and  3HC1). 

Prob.  16.  —  Determination  of  Hydrolysis  by  Distribution  Experi- 
ments. —  A  0.05  volume-formal  solution  of  Na2NH4PO4  in  water  is 
found  to  be  in  equilibrium  with  a  0.00173  volume-formal  solution  of 
NH3  in  chloroform  at  18°.  Calculate  the  hydrolysis  of  the  salt  into 
NH4OH  and  Na+2HPO4=.  The  distribution-ratio  for  NH3  between 
water  and  chloroform  at  18°  is  27.5. 

The  equilibrium-constant  of  a  hydrolytic  reaction  is  called  the 
hydrolysis-constant  #H  of  the  salt.  Thus  the  hydrolysis-constants  of 


156  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

potassium  cyanide  and  ammonium  cyanide  are  given  by  the  expres- 
sions: 

(NH4OH)  (HCN) 


_      yOH-)  (HCN)  (NH4OH)  (HCN 

(CN-)  H~   (NH4+)  (CN-) 

In  evaluating  these  constants  the  largely  ionized  substances  may  as 
an  approximation  be  regarded  as  completely  ionized,  as  is  to  be  done  in 
the  problems  throughout  this  chapter.  It  will  be  noted  that  the  error 
made  in  assuming  complete  ionization  of  the  potassium  cyanide  and 
potassium  hydroxide  is  largely  eliminated,  since  these  two  substances 
are  probably  not  far  from  equally  ionized  and  since  their  concen- 
trations occur  as  a  ratio  in  the  expression  for  the  hydrolysis-constant. 
A  similar  compensation  of  the  errors  arising  from  the  assumption  of 
complete  ionization  is  frequent  in  mass-action  formulations.  It  will 
be  seen,  however,  that  such  a  compensation  does  not  take  place  in  the 
hydrolysis-constant  of  the  ammonium  cyanide  when  this  is  evaluated 
under  the  assumption  that  it  is  completely  ionized;  but  even  in  this 
case,  when  this  constant  is  used  conversely  for  calculating  hydrolysis 
(as  in  Prob.  ijc)  the  error  is  small  so  long  as  the  concentration  of  the 
salt  is  not  greatly  different  from  that  for  which  the  constant  was 
evaluated. 

Just  as  the  equilibrium-constant  of  a  metathetical  gas  reaction  is 
determined  by  the  dissociation-constants  of  the  compounds  present 
(Art.  101),  so  the  equilibrium-constant  for  any  reaction  between 
ionized  substances  in  solution  is  determined  by  the  ionization-con- 
stants  of  the  partially  ionized  substances  involved.  Accordingly,  the 
hydrolysis-constant  of  a  salt  is  determined  (as  shown  in  Prob.  18) 
by  the  ionization-constant  Kw  of  water  and  the  ionization-constants 
KA  and  K*  of  the  slightly  ionized  acid  and  base  produced.  The 
hydrolysis  of  salts  can  therefore  be  calculated  from  these  ionization- 
constants  (as  in  Prob.  19);  and  conversely,  these  ionization-constants 
can  be  derived  from  experimentally  determined  values  of  the  hydrolysis 
(as  in  Prob.  20). 

Prob.  17.  —  The  Hydrolysis-Constant  in  Relation  to  the  Hydrolysis.  — 
a.  Calculate  from  the  data  in  the  above  text  the  value  of  the  hydrolysis- 
constant  at  25°  for  KCN,  and  for  NH4CN.  b.  Formulate  a  general 
expression  showing  how  the  hydrolysis  h  of  the  salt  varies  with  its 
concentration  c  in  the  case  of  KCN,  and  of  NH4CN.  c.  Calculate 
the  hydrolysis  of  NH4CN  in  a  solution  o.oi  formal  in  NH4CN  and  o.oi 
formal  in  NH4OH  at  25°. 


REACTIONS  BETWEEN  SOLUTES  157 

Prob.  18.  —  The  Hydrolysis-Constant  in  Relation  to  the  lonization- 
Constants.  —  a.  Calculate  the  values  of  the  hydrolysis-constant  at  25° 
for  KCN  and  for  NH4CN  from  the  ionization-constants  involved. 
b.  Formulate  general  expressions  for  the  relation  between  the  hydrolysis- 
constant  KB  of  salts,  like  KCN  and  NH4CN,  and  the  ionization-con- 
stants Kw,  KA,  K*. 

Prob.  19.  —  Calculation  of  Hydrolysis  from  the  lonization-Constants.  — 
a.  Calculate  the  concentrations  of  NH4OH,  H+,  and  OH~,  in  a  o.i  f. 
NH4C1  solution  at  25°  from  the  ionization-constants.  b.  Calculate 
the  hydrolysis  of  NH4C2H3O2  in  0.025  formal  solution  at  25°  from  the 
ionization-constants. 

Prob.  20.  —  Calculation  of  the  lonization  of  Water  from  the  Hydrolysis 
of  Salts.  —  Calculate  the  ionization-constant  of  water  at  100°  from  the 
hydrolysis  of  NH4C2H3O2  as  determined  in  Prob.  14  and  from  the 
ionization-constants  of  NH4OH  and  HC2H3O2  at  100°,  which  have  been 
found  by  conductance  measurements  to  be  i.4Xio~6and  i.iXio~B, 
respectively. 

106.  lonization  of  Dibasic  Acids  and  Their  Acid  Salts.  —  Polybasic 
acids  ionize  in  stages;  thus,  a  dibasic  acid  H^A  ionizes  according  to 
the  equations  H2A=H++HA~  and  HA-=H++A=.  The  equi- 
librium-constants Ki  and  Kz  of  these  reactions  are  called  the  ionization- 
constants  for  the  first  hydrogen  and  for  the  second  hydrogen,  respec- 
tively. The  values  of  Kz  are  commonly  much  smaller,  than  those  of 
Ki.  The  values  at  25°  of  the  two  constants  for  some  important  acids 
are  given  in  the  following  table. 

IONIZATION-CONSTANTS  OF  POLYBASIC  ACIDS 

Acid  Ki  &  K3 

H2SO3  1.7X10^  5Xio~« 

H3PO4  iXicr2  2X10^                icr" 

I^CJHA  (tartaric)  9-7  X  io~4  4  X  icT5  ' 

H2CO8  SXio-7  6X10-" 

H2S 


*From  the  two  ionization-constants  of  a  dibasic  acid  and  its  formal 
concentration  c  the  concentrations  of  the  various  substances  H2A, 
HA~,  A=,  andH+  present  in  its  solution  can  be  calculated.  Insolving 
such  mass-action  problems  involving  many  substances,  the  best  plan 
is  to  formulate  first  the  equilibrium  equations  that  must  be  satis- 
fied —  thus,  in  this  case  the  equations  (H+)  (HA~)  =  Ki  (H2A)  and 
(H+)  (A*=)  =  K2  (HA~).  The  next  step  is  to  formulate  the  so-called 
condition  equations,  which  sum  up  the  concentrations  of  the  separate 
forms  in  which  an  element  or  other  constituent  exists  in  the  solution. 


158  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

Thus,  in  this  case,  for  the  totalmolal  concentration  S(A)  of  the  con- 
stituent A,  and  for  that  S(H)  of  the  hydrogen,  we  have: 

2(A)  =c, 
=  S(H)  =  2c. 

(It  will  be  noted  that  one  of  these  equations  might  be  replaced  by  a 
simpler  condition-equation  of  another  kind;  namely,  by  the  equation, 
(H+)  =  (HA-)+2(A~),  which  expresses  the  fact  that  positive  and 
negative  ions  must  be  present  in  equivalent  quantities.)  We  now 
have  four  independent  equations  containing  four  unknown  concen- 
trations. An  exact  algebraic  solution  of  these  equations  is  therefore 
possible.  But,  as  such  a  solution  is  often  very  complicated,  it  is 
advisable  to  try  first  to  simplify  the  condition-equations,  which  can 
often  be  done  by  neglecting  in  them  the  concentration  of  some  sub- 
stance which  is  small  in  comparison  with  the  concentrations  of  each  of 
the  other  substances  occurring  in  the  same  equation.  Thus,  hi  this 
case  the  concentration  (A=)  is  small  compared  with  the  concentration 
H^A  because  even  the  first  hydrogen  is  split  off  to  only  a  moderate 
extent,  and  it  is  small  compared  with  the  concentration  (HA~)  for  the 
reasons  that  K%  is  small  in  comparison  with  Ki  and  that  the  hydrogen- 
ion  produced  by  the  ionization  of  the  H2A  into  H+  and  HA~  further 
decreases  by  the  common-ion  effect  the  ionization  of  the  HA~~.  It  is 
always  well  to  test  the  correctness  of  such  simplifying  assumptions 
by  subsequent  calculation  of  the  quantity  neglected. 

*Prob.  21.  —  Concentrations  of  the  Substances  Present  in  the  Solution 
of  a  Dibasic  Acid.  —  From  the  ionization-constants  given  in  the  table 
above,  calculate  by  the  method  just  described  the  molal  concentration 
of  each  of  the  substances  present  in  a  o.oi  formal  solution  of  tartaric 
acid  at  25°. 

*Prob.  22.  —  Determination  of  the  lonization-Constant  for  the  Second 
Hydrogen  by  Reaction-Rate  Measurements.  —  By  measuring  the  conduct- 
ance of  tartaric  acid  in  0.06-0.01  formal  solution,  where  the  ionization 
of  the  second  hydrogen  is  negligible,  the  ionization-constant  for  the 
first  hydrogen  of  the  acid  has  been  found  to  be  0.00097  at  25°.  The 
hydrogen-ion  concentration  in  a  o.i  formal  solution  of  sodium  hydro- 
gen tartrate,  NaHA,  has  been  shown  by  its  catalytic  effect  (in  Prob.  22 
of  Art.  93)  to  be  0.00020  molal.  From  these  data  calculate  the  con- 
centrations (A=),  (HA~~)  and  (H2A),  and  the  ionization-constant  for  the 
second  hydrogen  of  tartartic  acid.  (In  this  case  no  simplification  of 
the  condition-equations  is  admissible,  other  than  the  assumption  that 
the  salts  NaHA  and  Na2A  are  completely  ionized.)  Tabulate  the 
molal  concentrations  of  all  the  substances  present  in  the  solution  of  the 
sodium  hydrogen  tartrate. 


REACTIONS  BETWEEN  SOLUTES 


159 


Determination  of  the  lonization-Constant  for  the  Second  or  Third 
Hydrogen  of  Acids  by  Distribution  Experiments.  — 

*Prob.  2j.  A  o.i  volume-formal  solution  of  sodium  hydrogen  succin- 
ate  in  water  is  found  by  experiment  to  be  in  equilibrium  at  25°  with  a 
0.00187  volume-formal  solution  of  succinic  acid  in  ether.  The  distribu- 
tion-ratio of  succinic  acid  between  water  and  ether  at  25°  is  7.5. 

a.  Find  the  concentration  of  unionized  succinic  acid  in  the  aqueous  solu- 
tion of  the  acid  salt.     b.  Calculate  the  ionization-constant  for  the 
second  hydrogen.    That  for  the  first  hydrogen  is  6.6Xio-6.    Neglect 
the  value  of  (H+)  in  the  condition-equation,    c.  Show  by  calculating  the 
value  of  (H>)  that  it  was  justifiable  to  neglect  it.    Ans.  b,  2.5  Xio-«. 

*Prob.  24.  Calculate  the  ionization-constant  at  18°  for  the  third 
hydrogen  of  H3PO4  (that  for  the  ion  HPO4=)  from  the  data  of  Prob.  16 
and  from  the  ionization-constants  of  water  and  of  ammonium  hydroxide, 
which  at  18°  are  0.5X10-14  and  i.7Xio~5,  respectively. 

107.  Displacement  of  One  Acid  or  Base  from  its  Salt  by  Another.  — 

One  of  the  most  important  types  of  equilibrium  in  aqueous  solution 
is  the  partial  displacement  of  one  acid  or  base  from  its  salt  by  another; 
for  example,  that  of  acetic  acid  from  sodium  acetate  by  formic  acid, 
or  that  of  ammonium  hydroxide  from  ammonium  chloride  by  sodium 
hydroxide. 

Before  the  mass-action  relations  involved  were  fully  understood, 
the  extent  of  the  displacement  was  taken  as  a  measure  of  the  relative 
strengths  of  different  acids  or  bases;  those  which  are  largely  displaced 
from  their  salts  being  called  weak  acids  or  bases,  and  those  which 
cause  such  displacement  being  called  strong  acids  or  bases.  It  is 
shown  by  Probs.  25-28  that  the  mass-action  law  and  ionic  theory 
give  a  comparatively  simple  explanation  of  this  phenomenon  in  the 
case  of  not  largely  ionized  univalent  acids  or  bases;  also  that  the 
relative  strengths  of  different  acids  or  bases,  as  shown  by  their  dis- 
placements, are  determined  by  their  ionization,  weak  ones  being 
those  which  are  slightly  ionized  and  strong  ones  those  which  are 
largely  ionized. 

Displacement  of  Acids  or  Bases  from  their  Salts.  — 
Prob.  25.  —  To  a  o.i  formal  solution  of  KNO2  is  added  at  25°  an  equal 
volume  of  a  o.i  formal  solution  of  acetic  acid.    a.  Calculate  the  frac- 
tion of  the  potassium  nitrite  that  is  converted  into  potassium  acetate. 

b.  Calculate  the  fraction  that  would  be  so  converted  if  the  acetic  acid 
solution  were  i.o  formal  (instead  of  o.i  formal). 

Prob.  26.  —  a.  For  the  general  case  expressed  by  the  equation  B  A  + 
HA'  =B+A'~+HA  where  the  solution  is  originally  c  formal  in  B+A~  and 
cr  formal  in  HA',  derive  an  expression  for  the  fraction  x  of  the  salt  B+A~ 


160  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

converted  into  the  salt  B+A'~,  assuming  that  the  ionization-constants  of 
both  acids  are  so  small  that  the  ionization  of  them  in  the  presence  of 
their  salts  is  negligible,  b.  Derive  from  this  expression  the  relation  be- 
tween the  ionization-constants  of  the  acids  and  the  fractions  of  the  basic 
constituent  that  are  combined  with  the  two  acidic  constituents  for  the 
case  that  c  =cr;  and  state  the  principle  fully  in  words. 

Prob.  27.  —  To  a  liter  of  o.i  formal  HC1  solution  is  added  at  25°  a  liter 
of  o.i  formal  NH4OH  solution  and  a  liter  of  o.i  formal  CH3NH3OH  solu- 
tion. Calculate  the  fraction  of  the  acid  which  combines  with  each  base. 

Prob.  28.  —  A  o.  i  formal  solution  of  acetic  acid  is  added  to  an  equal 
volume  of  o.i  formal  NaHCO3  solution  at  25°,  the  carbon  dioxide  pro- 
duced being  kept  above  the  solution  at  a  pressure  of  i  atm.  a.  Calcu- 
late the  concentrations  of  the  two  salts  and  two  acids  in  the  resulting 
solution,  taking  0.0338  as  the  solubility-constant  (Art.  39)  for  CO2  in 
water  at  25°.  b.  Repeat  the  calculation  for  the  case  where  no  carbon 
dioxide  is  allowed  to  escape  from  the  solution;  and  tabulate  the  two 
results. 

The  displacement  of  acids  and  bases  from  their  salts  can  often  be 
determined  by  measuring  the  heat  effect  or  the  change  in  volume  (as 
in  Prob.  30)  attending  the  addition  of  one  acid  to  the  salt  of  the  other 
acid,  or  of  one  base  to  the  salt  of  the  other  base.  In  cases  where  one 
of  the  acids  has  a  color  different  from  that  of  its  salt  (as  in  Prob.  29), 
the  displacement  can  be  determined  by  measuring  the  attendant 
color  change. 

Prob.  29.  —  Determination  of  Displacement  by  Color  Measurements.  — 
Violuric  acid  is  a  slightly  ionized  acid  whose  solution  is  colorless,  while 
solutions  of  its  salts  have  a  violet  color  which  increases  proportionately 
with  the  quantity  of  salt.  25  ccm.  of  a  o.oi  formal  solution  of  violuric 
acid  are  placed  in  each  of  two  tubes  of  the  same  diameter;  to  one  tube 
are  added  25  ccm.  of  a  o.oi  formal  solution  of  potassium  acetate;  and 
to  the  other  tube  is  added  o.oi  formal  KOH  solution  until  on  looking 
down  through  the  tubes  the  colors  are  seen  to  be  the  same,  13.75  ccm- 
of  the  KOH  solution  being  required,  a.  Calculate  the  fraction  of  the 
violuric  acid  which  exists  in  the  first  mixture  in  the  form  of  its  salt. 
b.  Calculate  the  ionizat ion-constant  of  the  violuric  acid. 

*  Prob.  30.  —  Determination  of  Di  placement  by  Volume  Measure- 
ments. —  When  1000  g.  of  a  solution  containing  iKOH  is  mixed  with 
1000  g.  of  a  solution  containing  iHC2H3O2  there  is  an  increase  of  volume 
of  9.52  ccm.  When  the  former  solution  is  mixed  with  1000  g.  of  a  solu- 
tion containing  iHCHO2  there  is  an  increase  of  volume  of  12.39  ccm. 
When  2000  g.  of  a  solution  containing  iKCHO2  is  mixed  with  1000  g. 
of  one  containing  iHC2H3O2  there  is  a  decrease  of  volume  of  0.74  ccm. 
a.  Determine  what  fraction  of  the  formic  acid  is  displaced  from  its  salt 
by  the  acetic  acid.  b.  Calculate  from  the  ionization-constants  the 
fraction  displaced,  and  compare  the  results.  Ans.  a,  0.26. 


REACTIONS  BETWEEN  SOLUTES  161 

108.  Neutralization-Indicators.  —  An  acid  neutralization-indicator 
(such  as  litmus,  paranitrophenol,  or  phenol  phthalein)  is  a  mixture  of 
two  isomeric  acids  (HIn'  and  HIn  '0  in  equilibrium  with  each  other, 
one  of  which  (HIn")  is  present  in  much  smaller  proportion,  but  is 
so  much  more  ionized  than  the  other  (HIn')  that  when  a  base  BOH 
is  added  the  salt  produced  is  almost  wholly  of  the  form  B  +In  "~.  The 
substances  HIn'  and  B+In'~  are  different  in  color  from  the  substances 
HIn"  and  B+In"~,  the  color  being  determined  only  by  the  molecular 
structure  of  the  group  In.  It  follows  from  these  conditions  that  the 
indicator  acid  changes  color  when  converted  into  its  salt.  For  ex- 
ample, phenolphthalein  exists  in  two  isomeric  forms  to  which  the 
following  molecular  structures  are  assigned: 


and    C6H4-C 

|\C6H4OH  |  \CeH4  =O 

CO  -  O  COOH 

The  first  is  the  (colorless)  form  HIn'  which  is  present  in  predominating 
proportion  in  acid  solution;  the  second  is  the  (red)  form  HIn",  which 
has  a  much  larger  ionization-constant,  and  hence  on  the  addition  of 
alkali  is  converted  into  its  (red)  salt  in  much  larger  proportion  than 
the  other  form. 

It  can  be  shown  by  the  mass-action  law  that  the  indicator  behaves 
as  if  it  were  a  single  acid  HIn  whose  salt  has  a  different  color  from  the 
acid  itself;  and  its  behavior  can  be  represented  by  a  single  constant 
called  the  indicator-constant  Ki.  Namely,  by  combining  the  two 
equilibrium  equations: 

(Kin")  (H+)(In-) 

(HIS7)-*  (Kin") 

there  results  the  equation 


(HInO 

Since  according  to  the  above  statements  (In'-)is  negligible  in  compari- 
son with  (In""),  and  (HIn")  is  negligible  in  comparison  with  (HIn'), 
the  total  indicator  acid  is  substantially  equal  to  (HIn'),  and  the  total 
indicator  salt  is  substantially  equal  to  (In"-).  Hence  in  the  above 
expression  the  ratio  (In"-)/(HIn')  may  be  replaced  by  the  ratio 


1  62  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

x/(i  —  x),  where  x  represents  the  fraction  of  the  total  quantity  of  the 
indicator  which  is  in  the  colored  form  that  exists  in  alkaline  solution. 
Making  this  substitution,  there  results  the  following  fundamental 
indicator  equation: 


The  relations  in  the  case  of  basic  indicators  (existing  almost  wholly 
in  the  two  differently  colored  forms  In'  OH  and  In"+  A~)  are  entirely 
similar  to  those  in  the  case  of  acid  indicators.  By  formulating  the 
mass-action  equations,  it  can,  in  fact,  be  shown  that  the  relation  just 
derived  between  the  hydrogen-ion  concentration  and  the  fraction  x 
of  the  indicator  transformed  holds  true  also  in  the  case  of  basic  indi- 
cators. A  basic  indicator  may  therefore  be  treated  as  an  acid  indi- 
cator by  adopting  as  the  value  of  its  indicator-constant  that  calculated 
by  the  above-given  indicator  equation;  and  it  will  be  so  treated  in  the 
following  problems. 

Prob.  31.  —  Determination  of  the  Indicator-Constant,  —  To  100  ccm. 
of  a  solution  o.i  normal  in  potassium  acetate  and  o.oi  normal  in  acetic 
acid  are  added  10  ccm.  of  a  o.oi  normal  solution  of  paranitrophenol. 
This  solution  is  found  to  have  the  same  yellow  color  as  a  solution  made 
by  adding  0.50  ccm.  of  o.oi  normal  paranitrophenol  solution  and  9.5 
ccm.  of  o.oi  normal  KOH  solution  to  100  ccm.  of  water,  a.  Calculate 
the  indicator-constant,  neglecting  the  quantity  of  the  acetic  acid  dis- 
placed from  its  salt  by  the  small  proportion  of  paranitrophenol  present. 
6.  Show  what  percentage  error  is  made  in  (H+),  and  therefore  in  Kxby 
neglecting  this  displaced  quantity. 

Prob.  32.  —  Determination  of  Small  Hydrogen-Ion  Concentrations  by 
Means  of  Indicators.  —  When  a  small  proportion  of  phenolphthalein 
(Ki  =  io~ltf)  is  added  to  a  o.i  formal  solution  of  NaHCO3  at  25°  the 
indicator  is  found  by  color  comparisons  to  be  6.0%  transformed  into  its 
salt.  Calculate  the  hydrogen-ion  and  hydroxide-ion  concentrations  in 
the  solution. 

It  is  evident  from  the  indicator  equation  that  in  the  titration  of  an 
acid  with  a  base  an  end-point  will  be  reached  when  the  hydrogen-ion 
concentration  (H+)  becomes  so  reduced  that  a  considerable  fraction  x 
of  the  indicator  is  transformed  into  its  salt;  also  that  the  observed 
end-point  will  be  correctly  located  only  if  this  transformation  occurs 
when  equivalent  quantities  of  acid  and  base  are  present,  and  that  the 
sharpness  of  the  end-point  will  depend  on  how  fast  the  hydrogen-ion 
concentration  changes  in  this  region.  Hence,  in  order  to  determine 
how  an  acid  of  any  known  ionization-constant  will  behave  on  titration 


REACTIONS  BETWEEN  SOLUTES 


163 


and  in  order  to  select  the  most  suitable  indicator,  it  is  necessary  to 
know  how  the  hydrogen-ion  concentration  varies  when  a  largely 
ionized  base  is  progressively  added.  The  following  table  contains 
data  of  this  kind,  calculated  as  shown  in  Prob.  33.  Namely,  it  shows 
the  hydrogen-ion  concentrations  prevailing  at  25°  in  solutions  of  acids 
of  ionization-constants  io~3,  io~,5  io~7,  and  io~9  when  the  ratio  B/A 
of  the  quantity  of  NaOH  added  to  the  quantity  of  the  acid  present 
has  the  values  given  in  the  first  column  (the  original  concentration  of 
the  acid  and  that  of  the  standard  base  being  0.2  normal). 

CHANGE  OF  THE  HYDROGEN-ION  CONCENTRATION  ON  NEUTRALIZATION 


Ratio 

Values  of  the  hydrogen-ion  concentration  for 

B/A 

KA  =  zo-3 

KA  =  /o-5 

KA  =  io-7 

K     =  jQ-9 

0.980 

2XIO-6 

2XIO-7 

2XIO-9 

2.4X10-" 

0.990 

iXio-6 

iXio-7 

iXio-9 

1.6  Xio-" 

0-995 

SXio-8 

SXio-8 

SXio-10 

1.2X10-" 

0.998 

2  XIO-6 

2X10-8 

2.4  Xio-10 

1.  1  Xio-" 

.000 

iX  to-8 

iXio~9 

i  Xio-10 

1.0X10-" 

.002 

SXio-11 

SXio-11 

4  Xio-11 

0.9X10-" 

.005 

2  XIO-11 

2  Xio-11 

2X10-" 

0.8X10-" 

.OIO 

i  Xio-11 

i  Xio-11 

i  Xio-11 

0.6  Xio-" 

.020 

5  Xio-12 

5  Xio-12 

5  Xio-12 

0.4X10-" 

These  numbers  are  also  the  values  of  the  hydroxide-ion  concentra- 
tion for  the  case  that  a  0.2  normal  solution  of  a  base  having  an 
ionization-constant  equal  to  lo"3,  io~5,  io~7,  or  icr9  is  titrated  with 
a  0.2  normal  solution  of  a  strong  acid  (like  HC1),  provided  the 
numbers  in  the  first  column  denote  the  ratio  (A/B)  of  the  quantity 
of  acid  to  the  quantity  of  base.  Thus,  when  the  ratio  A/B  is  0.98, 
the  value  of  (OH~)  is  2  X  lo"5  for  a  base  for  which  K  =  lo"8. 

Titration  of  Acids  and  Bases.  — 

Prob.  jj.  —  100  ccm.  of  a  0.2  normal  solution  of  an  acid  whose  ioni- 
zation-constant is  io-5  are  titrated  at  25°  with  0.2  normal  KOH  solu- 
tion. Calculate  the  hydrogen-ion  concentration  (H+)  in  the  mixture 
when  99.0,  99.5,  99.8,  100.0,  100.2,  100.5,  and  101.0  ccm.  of  the  KOH 
solution  have  been  added,  using  i  X  io~14  as  the  ionization-constant 
of  water. 

Prob.  34.  —  a.  Plot  the  common  logarithms  of  the  values  of  the 
hydrogen-ion  concentrations  given  in  the  preceding  table  as  ordinates 
against  the  corresponding  values  of  the  ratio  of  base  to  acid  as  abscissas, 
for  each  of  the  four  acids,  b.  On  the  right-hand  side  of  the  same 
diagram  write  in  a  scale  of  values  of  logio(OH~)  corresponding  to  the 
values  of  logio(H+)  on  the  left-hand  side;  and  at  the  top  of  the  diagram 
write  in  a  scale  of  ratios  of  acid  to  base  (A/B)  corresponding  to  the 


1 64  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

scale  of  ratios  of  base  to  acid  at  the  bottom  (thus,  B/A=o.g8  corre- 
sponds to  A/B  =  1.02).  Now  make  plots  on  the  same  diagram,  showing 
how  (H+)  or  (OH~)  varies  in  titrating  bases  of  ionization-constants 
io~3,  io-6,  io-7,  and  io~9  with  HC1  at  25°,  similar  in  all  respects  to  the 
plots  previously  made  for  the  four  acids. 

Prob.  35.  —  a.  From  a  study  of  the  diagram  of  Prob.  34  tabulate 
the  values  between  which  the  indicator-constant  must  lie  in  order  that 
the  titration  of  each  of  the  four  acids  and  four  bases  may  be  correct 
within  0.2%,  assuming  that  the  indicator  is  9%  transformed,  b.  Show 
from  the  plot  what  percentage  error  would  be  made  in  using  phenol 
phthalein  (Ki  =  io~~10)  hi  titrating  an  acid  for  which  KA  —  io~5  when 
the  fraction  of  the  indicator  transformed  is  i%,  9%,  and  50%;  also 
in  titrating  an  acid  for  which  J£A  =  io-7  when  the  fraction  of  the  in- 
dicator transformed  is  9%  and  50%.  c.  If  the  acid  for  which  JK"A  =  io-9 
were  titrated  with  the  aid  of  an  indicator  for  which  KI  =  io-12  (which 
is  not  far  from  the  value  for  trinitrobenzene),  what  error  would  be 
made  when  the  fraction  of  the  indicator  transformed  is  5%,  9%,  and 
I5%?  (Note  that  hi  a  titration  carried  out  in  the  usual  way  the 
fraction  transformed  is  not  determined  more  closely  than  this.  Note 
also  that  an  error  in  the  assumed  value  of  the  indicator-constant 
would  affect  the  results  in  the  same  way  as  a  variation  in  the  fraction 
transformed.) 

Prob.  36.  —  Calculate  the  value  of  (H+)  at  the  end-point  in  titrating 
with  phenolphthalein  (KI  =  io~10}  when  the  fraction  x  of  it  trans- 
formed is  5%  and  20%;  with  rosolic  acid  (JfiTI  =  io~8)  when  x  is  5% 
and  20%;  with  paranitrophenol  (^i  =  io~7)  when  x  is  i%  and  20%; 
and  with  methyl  orange  (KI  =  5XicT*')  when  x  is  80%  and  95%  (these 
being  about  the  ranges  practicable  hi  a  titration).  Draw  in  on  the 
diagram  made  in  Prob.  34  horizontal  lines  representing  these  limiting 
values  of  (H+)  for  the  four  indicators.  Letter  the  curves  and  lines  on 
the  diagram  so  as  to  show  what  each  represents. 

Prob.  37.  —  With  the  aid  of  the  diagram  show  which  of  these  indi- 
cators would  give  a  result  accurate  within  0.2-0.3%  in  titrating 
a,  NH4OH  with  HC1;  b,  HNO2  with  KOH;  c,  aniline  (C6H5NH3OH) 
with  HC1. 

Prob.  38.  —  Titration  of  the  Successive  Hydrogens  of  Poly  basic  Acids.  — 
A  solution  0.2  formal  in  H3P04  is  titrated  with  0.2  formal  NaOH  solu- 
tion. With  the  aid  of  the  ionization-constants,  calculate  the  H+  ion 
concentration  after  the  addition  of  enough  base  to  be  equivalent,  a,  to 
99%  of  the  first  hydrogen  of  the  acid,  b,  to  i%  of  the  second  hydrogen, 
c,  to  99%  of  the  second  hydrogen,  and  d,  to  i%  of  the  third  hydrogen. 
e.  By  reference  to  the  diagram  determine  what  indicators  could  be  used 
for  titrating  the  first  hydrogen  and  the  second  hydrogen  of  the  acid. 


REACTIONS  INVOLVING  SOLIDS  165 

IV.    THE  MASS-ACTION  LAW  FOR  REACTIONS  INVOLVING  SOLID  PHASES 

109.  Form  of  the  Mass-Action  Expression.  —  When  a  substance 
present  as  a  solid  phase  is  involved  in  a  reaction  with  gaseous  sub- 
stances at  small  pressures,  that  substance  has  in  the  gaseous  phase  of 
all  equilibrium  mixtures  at  any  definite  temperature  the  same  pressure, 
namely,  one  equal  to  the  vapor-pressure  of  the  solid  substance.     Simi- 
larly, when  a  substance  present  as  a  solid  phase  is  involved  in  a  re- 
action with  dissolved  substances  at  small  concentrations,  it  has  in 
the  liquid  phase  of  all  equilibrium  mixtures  at  any  definite  tempera- 
ture the  same  concentration,  namely,  one  equal  to  its  concentration 
in  a  solution  saturated  with  the  solid  substance  and  containing  no 
other  solutes.    Hence  the  pressure  in  the  gaseous  phase  or  the  con- 
centration in  the  liquid  phase  of  any  substances  which  are  also  present 
as  solid  phases  may  be  left  out  in  formulating  the  mass-action  expres- 
sion, their  constant  pressures  or  concentrations  being  understood  to  be 
included  in  the  equilibrium-constant.    Thus,  p*2/p&o  =  K  is  the  mass- 
action  expression  for  the  reaction  Fe(s)+H2O(g)  =  FeO(s)+H2(g). 

As  here  illustrated,  solid,  liquid,  and  gaseous  substances  occurring 
in  chemical  equations  are  indicated  by  attaching  to  their  formulas  (s), 
(1),  and  (g),  respectively,  when  it  is  important  to  show  the  state  of 
aggregation.  Substances  in  solution  are  written  either  without  any 
such  addition  to  their  formulas  or  with  the  addition  of  a  parenthesis 
showing  the  concentration  or  composition;  for  example,  NH3  (o.i  f.), 
H+Cl-(in  100  H2O). 

110.  Reactions  involving  Solid  and  Gaseous  Substances.  —  The 
simplest  type  of  the  reactions  involving  solid  and  gaseous  substances 
is  that  in  which  only  one  substance  is  present  in  appreciable  quantity 
in  the  gaseous  phase.    Examples  of  this  type  are: 

2Ag20(s)  =  4Ag(s)+02(g);  CaCO3(s)  =CaO(s)+C02(g); 
CaS04.2H2O(s)  =CaSO4(s)  +  2H20(g). 

The  mass-action  expression  for  this  case  is  simply  p  =  K,  which  sig- 
nifies that  at  any  definite  temperature  there  is  only  one  pressure  of 
the  gas  at  which  there  can  be  equilibrium.  This  pressure  is  called 
the  dissociation-pressure  of  the  substance  undergoing  decomposition 
(thus  of  the  silver  oxide,  the  calcium  carbonate,  or  the  gypsum).  If 
the  pressure  of  the  gas  is  kept  larger  than  this  pressure,  the  reaction 
takes  place  completely  in  one  direction,  with  the  result  that  the  gas 
is  entirely  absorbed;  and  if  the  pressure  is  kept  smaller,  the  reaction 


1 66  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

takes  place  completely  in  the  other  direction,  with  the  result  that  the 
dissociating  substance  completely  decomposes.  This  important  char- 
acteristic of  reactions  of  this  type  will  be  fully  considered  in  Art.  119. 
Other  types  of  reactions,  involving  equilibrium  between  solid 
substances  and  two  or  more  gaseous  substances,  are  illustrated  by  the 
following  problems. 

Dissociation  of  Solid  Substances  Producing  Two  Gaseous  Substances.  — 

Prob.  39.  —  When  solid  NH4SH  is  -placed  in  a  vacuous  space  at  25° 
a  pressure  of  500  mm.  is  developed,  owing  to  the  dissociation  of  the 
salt,  which  is  itself  not  appreciably  volatile,  into  NH3  and  H2S.  If 
solid  NH4SH  were  introduced  into  a  space  which  already  contained 
H2S  at  a  pressure  of  300  mm.,  what  increase  of  pressure  would  result? 

Prob.  40.  —  The  equilibrium-constant  (expressed  in  atmospheres)  of 
the  reaction  2NaHCO3(s)  =Na2CO3(s)+CO2(g)+H20(g)  at  100°  is 
0.23.  A  current  of  moist  carbon  dioxide  is  passed  at  i  atm.  over  solid 
sodium  hydrogen  carbonate  (in  order  to  free  it  from  adhering  water). 
How  great  must  the  mol-fraction  of  the  water  in  the  gas  be  to  prevent 
decomposition  of  the  salt? 

Prob.  41.  —  When  solid  mercuric  oxide  is  heated  in  an  evacuated  tube 
to  357°  in  the  vapor  of  mercury  boiling  at  i  atm.,  a  pressure  of  70  mm. 
is  produced  in  the  tube  owing  to  dissociation  of  the  oxide  into  mercury 
vapor  and  oxygen.  Assuming  that  a  mixture  of  solid  mercuric  oxide 
and  liquid  mercury  is  heated  in  the  same  way,  calculate  the  partial 
pressure  of  the  oxygen  in  the  tube.  Ans.  0.088- mm. 

Prob.  42.  —  Equilibrium  of  the  Producer  Gas  Reaction.  —  In  one  stage 
of  the  manufacture  of  producer  gas  a  current  of  air  at  i  atm.  is  passed 
through  coke  at  a  high  temperature.  Calculate  the  mol-fractions  of 
CO2  and  of  CO  in  the  gas  issuing  at  1000°,  assuming  equilibrium  to  be 
reached.  The  equilibrium-constant  for  the  reaction  C(s)+C02(g)  = 
2CO(g)  at  1000°  is  140. 

Dissociation-Pressure  in  Relation  to  Equilibrium-Constants.  — 
Prob.  43.  —  At  1 1 20°  the  gaseous  mixture  in  equilibrium  with  iron 
and  ferrous  oxide  consists  of  54  mol-percent  of  H2  and  46  mol-percent 
of  H2O.    Calculate  the  dissociation-pressure  of  ferrous  oxide,  referring 
to  Prob.  8  for  the  additional  data  needed. 

Prob.  44.  —  In  the  blast-furnace  process  iron  is  reduced  by  the  reac- 
tion FeO(s)  +C0(g)  =Fe(s)  +  CO2(g).     Calculate  the  least  quantity  of 
I        carbon  monoxide  that  could  reduce  one  formula-weight  of  FeO  at  1120°, 
using  the  dissociation-pressure  of  ferrous  oxide  found  in  Prob.  43. 
111.   Solubility  of  Unionized  and  Slightly  Ionized  Substances.  — 
The  fundamental  principle  determining  the  solubility  of  a  unionized 
or  slightly  ionized  substance  in  the  presence  of  other  solutes  is  that 
the  concentration  of  this  substance,  or  of  its  unionized  part  if  its 


REACTIONS  INVOLVING  SOLIDS  167 

ionization  is  not  negligible,  has  the  same  value  at  any  definite  tempera- 
ture in  any  solution  saturated  with  respect  to  this  substance,  whatever 
other  solutes  may  be  present  at  small  concentrations.  The  application 
of  this  principle  is  illustrated  by  the  following  problems. 

Solubility-Increase  Due  to  Complex  Formation.  — 

Prob.  45.  —  At  25°  the  solubility  of  iodine  in  water  is  0.0013  molal, 
and  its  solubility  in  o.i  formal  KI  solution  is  0.0517  molal,  the  increase 
being  due  to  the  reaction  K+I~+I2(s)  =K+I3~.  Calculate  its  solubility 
in  o.oi  formal  KI  solution  at  25°. 

Prob.  46  —  Calculate  the  solubility  of  HgCl2  in  o.i  formal  NaCl  solu- 
tion at  25°  from  the  results  of  the  distribution  experiments  of  Prob.  44 
of  Art.  45  and  from  the  fact  that  its  solubility  in  water  at  25°  is  0.263 
molal. 

112.  Solubility  of  Largely  Ionized  Substances.  —  The  mass-action 
law  evidently  requires,  in  all  dilute  solutions  saturated  with  the  same 
solid  substance,  that  the  concentration  of  the  unionized  substance 
have  the  same  value,  and  also  that  the  product  of  the  ion-concentra- 
tions raised  to  the  appropriate  power  have  the  same  value,  whatever 
other  substances  may  be  present  (at  small  concentrations).  Thus  at 
any  definite  temperature  in  any  aqueous  solution  saturated  with 
silver  sulfate,  whether  it  is  present  alone  or  with  another  solute,  such 
as  silver  nitrate  or  ammonium  hydroxide,  the  first  of  these  principles 
requires  that  (Ag2S04)  have  a  definite  value;  and  the  second  principle 
requires  that  the  product  (Ag+)2X(SO4=)  also  have  a  definite  value. 
This  saturation  value  of  the  ion-concentration  product  is  commonly 
called  the  solubility-product]  and  the  principle  expressing  its  con- 
stancy is  called  the  solubility-product  principle. 

The  principle  relating  to  the  concentration  of  the  unionized  part 
cannot  be  applied  to  largely  ionized  substances;  for  the  degree  of 
ionization  of  such  substances  is  so  imperfectly  known  that  it  is  not 
possible  to  estimate  even  roughly  the  concentration  of  the  unionized 
part.  In  the  case  of  largely  ionized  substances  the  solubility-product 
principle  is  therefore  employed;  but  even  this  gives  only  approximate 
results,  since,  as  shown  in  Art.  103,  ions  deviate  considerably  from  the 
laws  of  perfect  solutions  even  at  fairly  small  concentrations.  Never- 
theless, the  solubility-product  principle,  supplemented  by  the  hy- 
potheses that  largely  ionized  substances  are  completely  ionized  and 
that  their  ions  act  as  perfect  solutes,  yields  roughly  approximate 
results  in  the  treatment  of  the  solubility  effects  of  uniunivalent  sub- 
stances up  to  moderate  concentrations,  such  as  0.1-0.2  formal,  as 


1 68  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

will  be  shown  in  the  following  problems  by  comparison  of  the  observed 
solubility  values  with  those  calculated  from  the  solubility  in  pure 
water.  The  effects  of  uniunivalent  substances  on  the  solubility  of 
unibivalent  substances,  and  the  effects  of  unibivalent  substances  on 
the  solubility  of  uniunivalent  substances,  are  also  in  roughly  approxi- 
mate agreement  with  the  requirements  of  these  principles.  Very 
large  deviations  are  met  with,  however,  in  evaluating  the  solubility 
of  one  unibivalent  substance  in  the  presence  of  another.  Thus  the 
effect  of  Ag+NO3~on  the  solubility  of  Ag+2SO4=,  or  the  effect  of 
Ca-H-Cl~2  on  the  solubility  of  T1+C1~,  may  be  estimated  by  these 
principles;  but  the  effect  of  Na+2SO4=  on  the  solubility  of  Ag+2SO4= 
is  very  different  from  that  calculated.  This  very  abnormal  effect 
exerted  by  bivalent  ions  may  arise  either  from  great  deviations  in 
their  behavior  from  that  of  perfect  solutes,  or  from  their  conversion 
into  intermediate  ions  (such  as  AgSO4~  in  the  case  of  Ag2SO4).  In 
view  of  this  great  divergence  it  is  not  practicable  to  predict  even 
approximately  the  effect  of  unibivalent  salts  on  the  solubility  of  one 
another,  except  at  very  small  concentrations  (for  example,  below  o.oi 
formal). 

An  exact  method  of  treatment  of  solubility  effects  in  the  case  of 
largely  ionized  substances  is  described  in  Art.  113. 

Solubility-Decrease  by  Substances  with  a  Common  Ion.  — 

Prob.  47.  —  The  solubility  of  AgBr03  in  water  at  25°  is  0.0081  formal. 
Calculate  its  solubility  in  a  solution  0.0085  formal  in  AgNO3,  and  com- 
pare the  result  with  the  value  0.0051  formal  experimentally  determined. 

Prob.  48.  —  The  solubility  of  KC1O4  in  water  at  25°  is  0.148  formal. 
Calculate  its  solubility  in  a  solution  o.ioo  formal  in  KC1.  Compare  the 
result  with  the  value  0.112  formal  experimentally  determined. 

Prob}  49.  —  a.  Derive  an  expression  for  the  solubility  s  of  AgCl  in  a 
dilute  NaCl  solution  of  concentration  c  in  terms  of  its  solubility  s0  in 
water,  b.  Calculate  the  ratio  S/SQ  for  c  =  s0,  for  c  -  2%  and  c  -  IOSQ. 

Prob.  50.  —  The  solubility  of  Ag2SO4  in  water  at  25°  is  0.0268  formal. 
Calculate  its  solubility,  a,  in  a  solution  0.050  formal  in  AgNO3,  b,  in  a 
solution  0.025  formal  in  K2SO4,  solving  in  each  case  the  cubic  equation 
by  trial,  c.  Tabulate  the  results  with  those  actually  determined, 
which  are  0.0142  and  0.0247,  respectively. 

Prob.  51.  — The  solubility  of  Mg(OH)2  in  water  at  18°  is  i.4Xio~4 
formal,  a.  Calculate  its  solubility  in  0.002  formal  NaOH  solution. 
b.  Calculate  its  solubility  in  o.ooi  formal  MgCl2  solution. 

Prob.  52.  —  Solubility-Increase  through  Metathesis.  —  The  solubility 
of  Mg(OH)2  in  water  at  18°  is  1.4X10"*  formal,  a.  Calculate  its 
solubility  in  0.002  formal  NH4C1  solution.  Neglect  the  concentration 


REACTIONS  INVOLVING  SOLIDS  169 

of  OH~  In  comparison  with  that  of  NH4OH.  b.  Calculate  the  ratio  of 
(OH~)  to  (NH4OH)  in  the  saturated  solution  in  order  to  test  the  simpli- 
fying assumption  above  made.  Ans.  a,  8.6Xio~4. 

Prob.  55.  —  Solubility-Increase  through  Complex-Formation.  —  The 
solubility  of  AgCl  in  water  at  25°  is  i.3oXio~5  formal.  Calculate  its 
solubility  in  o.i  formal  NH3  solution.  There  is  formed  a  complex  ion 
by  the  reaction  Ag+-f  2NH3=Ag(NH3)2+,  its  equilibrium-constant 
(commonly  called  the  complex-constant)  having  the  value  i.4Xio7. 
Ans.  0.0044. 

Conversion  of  One  Solid  Substance  into  Another.  — 

Prob.  54.  —  The  solubility  of  silver  thiocyanate  is  i.2Xio~6  formal 
and  that  of  silver  bromide  is  0.7  X  io~6  formal  at  25°.  a.  Calculate  the 
equilibrium-constant  of  the  reaction  AgSCN(s)+K+Br~=AgBr(s)-f 
K+SCN~  in  dilute  solution,  b.  If  8.3  g.  of  solid  AgSCN  are  treated 
with  200  ccm.  of  o.i  formal  KBr  solution,  what  proportion  of  the  silver 
salt  is  converted  into  bromide?  c.  What  volume  of  the  o.i  formal 
KBr  solution  would  convert  the  solid  AgSCN  completely  into  AgBr? 
d.  With  what  mixtures  of  potassium  thiocyanate  and  potassium  bromide 
could  the  silver  thiocyanate  be  treated  without  any  change  taking 
place? 

Prob.  55.  —  Determine  the  ratio  of  carbonate  to  hydroxide  in  the 
solution  obtained  by  digesting  at  25°  a  o.i  formal  Na2C03  solution  with 
excess  of  solid  Ca(OH)2  (as  in  the  technical  process  of  causticizing  soda). 
The  solubility  of  calcium  hydroxide  is  0.020  formal,  and  the  product 
(Ca++)X(C03=)  has  the  value  3Xio~9  in  water  saturated  with  cal- 
cium carbonate. 

*113.   The  Mass-Action  of  Imperfect  Solutes.     The  Concept  of 

Activity.  —  The  mass-action  law  is  a  limiting  law  rigorously  exact 
only  for  perfect  gases  or  perfect  solutes,  but  holding  true  with  reason- 
able accuracy  in  the  case  of  most  gases  at  moderate  pressures  (such  as 
i  to  5  atmospheres),  and  in  the  case  of  solutes  with  electrically  neutral 
molecules  up  to  moderate  concentrations  (such  as  i  molal),  but  show- 
ing large  deviations  in  the  case  of  ions  even  at  small  concentrations 
(such  as  o.i  molal).  For  the  mass-action  and  thermodynamic 
treatment  of  solutes  at  concentrations  larger  than  those  at  which  they 
can  be  regarded  as  perfect  solutes,  it  has  been  found  convenient  to 
introduce  a  new  concept,  which  will  now  be  described. 

The  mass-action  of  a  perfect  gas  or  perfect  solute,  as  the  mass- 
action  law  states,  is  represented  by  its  concentration;  but  when  a 
chemical  substance  is  not  a  perfect  gas  or  solute  or  when  its  concen- 
tration is  unknown,  its  mass-action  is  expressed  by  the  term  activity  (a), 
by  which  is  meant  that  quantity  which,  when  substituted  for  the 
concentration  of  the  substance  in  mass-action  equations,  expresses 


170  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

its  effect  in  determining  the  equilibrium.  Hence  the  activity  of  a 
substance  represents  its  effective  concentration  from  a  mass-action 
viewpoint;  and  the  factor  by  which  theactual  concentration  c  must 
be  multiplied  to  give  the  activity  is  called  the  activity-coefficient  a,  that 
is,  a  =  a  c,  or  a  =  a/c. 

In  accordance  with  these  general  definitions,  the  most  obvious 
method  of  determining  the  relative  activities  of  a  substance  in  two 
solutions  of  different  concentrations,  or  in  general  in  any  two  phases, 
is  to  find  its  concentrations  hi  gaseous  phases  in  equilibrium  with  the 
two  solutions,  or  in  general  with  the  two  phases;  for  the  activity  of  a 
perfect  gas  is  placed  equal  to  its  concentration,  and  the  activity  of  a 
gas  at  small  pressure  can  ordinarily  be  considered  to  be  substantially 
equal  to  its  concentration.  This  method  affords  a  definition  of 
activity,  which  is  more  specific  and  quantitative  than  the  general 
statement  in  regard  to  it  made  above.  Namely,  the  ratio  of  the 
activities  of  a  substance  hi  two  phases  is  equal  to  the  ratio  of  its  vapor- 
pressures  in  the  two  phases,  provided  the  vapor  can  be  regarded  as 
perfect  gas.  That  is,  ai/O2=pi/pz-  For  example,  the  ratio  of  the 
activities  of  unionized  hydrochloric  acid  in  its  10  formal  and  7  formal 
aqueous  solutions  at  25°  is  30.0,  since  the  partial  vapor-pressures  of 
the  hydrochloric  acid  in  those  solutions  have  been  found  to  be  4.2 
and  0.14  mm.  of  mercury.  This  quantity  is  also  the  ratio  of  the 
products  of  the  activities  of  hydrogen-ion  and  chloride-ion  in  the  two 
solutions,  since  these  products  are  by  definition  the  quantities  that 
must  be  substituted  in  the  mass-action  expression  for  the  equilibrium 
of  the  reaction  HC1=H++C1~.  These  statements  are  expressed  by 
the  equations: 

(0HCl)l        (flH* 


Calculation  of  Activity  Ratios  from  Vapor-Pressure.  — 
Prob.  56.  —  a.  Calculate  the  relative  activities  of  NH3  and  H2O  in  i.o 
formal  and  in  o.i  formal  solution  at  25°  from  the  facts  that  the  vapor- 
pressures  of  NH3  in  these  solutions  are  13.52  mm.  and  1.334  mm.,  and 
that  the  vapor-pressures  of  H2O  are  those  required  by  Raoult's  law. 
b.  Calculate  the  ratio  of  the  activities  of  the  NH4OH  in  these  two  solu- 
tions; also  the  ratio  in  each  solution  of  the  activity  of  the  NH4OH  to 
that  of  the  NH3.  c.  Calculate  the  ratio  of  the  products  of  the  activities 
of  the  NH4+  and  OH~  ions  in  the  two  solutions. 

Prob.  57.  —  a.  Calculate  the  ratio  of  the  activities  of  H2O  in  pure 
water  and  in  i  formal  NaCl  solution  at  o°,  whose  mol-number  (Art.  59) 


REACTIONS  INVOLVING  SOLIDS 


171 


is  1.79,  as  determined  from  its  freezing-point,  b.  Find  also  the  ratio 
of  the  product  of  the  activities  of  the  H+  and  OH~  ions  in  pure  water 
to  the  product  in  i  formal  NaCl  solution  at  o°. 

This  direct  method  is  of  very  limited  applicability  to  largely  ionized 
substances,  since  they  seldom  have  appreciable  vapor-pressures. 
There  is,  however,  a  simple  thermodynamic  relation  between  activity 
and  electromotive  force  (considered  in  Arts.  146  and  147)  which 
furnishes  a  means  of  determining  the  change  in  the  product  of  the 
activities  of  the  ions  with  the  concentration  of  the  salt;  and,  by 
assuming  that  at  very  small  concentrations  the  activity  of  the  ions  is 
equal  to  the  concentration  of  the  ion-constituent,  absolute  values  of 
the  product  of  the  activities  of  the  ions  at  various  concentrations  of 
the  substance  are  obtained.  The  change  in  the  activity  of  solutes 
with  the  concentration  can  also  be  derived  from  the  freezing-point 
lowering  with  the  aid  of  thermodynamic  relations,  and  from  the  effect 
of  salts  on  the  solubility  of  one  another. 

A  number  of  substances  have  already  been  studied  by  these  methods. 
The  following  table  contains  some  typical  values  of  the  activity- 
coefficient  at  25°.  (These  values  are  in  reality  the  square-root  of 
the  product  of  the  activity-coefficients  of  the  two  ions  of  the  substance, 
or  the  geometrical  mean  of  the  activity-coefficients  of  the  two  ions; 
for  it  is  only  this  product  that  can  be  determined.)  These  values  are 
so  expressed  as  to  be  the  quantity  by  which  the  concentration  of  the 
substance  must  be  multiplied  to  give  the  activity  of  the  ions.  For 
comparison  the  values  of  the  conductance- viscosity  ratio  are  also  given 
in  the  table.  Potassium  chloride  and  ammonium  chloride  have  been 
found  to  have  substantially  the  same  activity-coefficients  and  the 
same  conductance-viscosity-ratios  as  sodium  chloride,  at  any  rate 
up  to  0.2  formal. 

ION-ACTIVITY  COEFFICIENTS 


I  on-  A  ctimty-Coefficients 

Conductance-  Viscosity-Ratios 

Formality 

NaCl 

LiCl 

HCl 

KOH 

KCl 

LiCl 

HCl 

KOH 

O.OOI 

0.98 

0.98 

0.99 

— 

0.98 

0.98 

0.99 

0.99 

O.OIO 

0.92 

0.92 

0.92 

0.92 

0.94 

0-93 

0.97 

0.96 

0.02O 

0.89 

0.89 

0.89 

0.89 

0-93 

0.92 

0.96 

0-95 

0.050 

0.84 

0.84 

0.86 

0.84 

0.89 

0.88 

0.94 

0.92 

O.IOO 

0.8o 

0.8o 

0.81 

0.80 

0.86 

0.85 

0.92 

0.91 

0.200 

0-75 

0.77 

0.78 

0-75 

0.83 

0.81 

0.91 

0.89 

0.500 

0.69 

o-7S 

0.76 

0-73 

0.78 

0.77 

0.89 

0.88 

I.OOO 

0.65 

0.78 

0.82 

0-75 

0.74 

0.74 

0.84 

0.88 

3.000 

0.70 

i.  20 

i-3S 

I.I 

— 

— 

— 

— 

172  EQUILIBRIUM  OF  CHEMICAL  CHANGES 

The  table  shows  that  with  increasing  concentration  the  ion-activity  - 
coefficient  of  all  four  substances  decreases  at  first  much  more  rapidly 
than  the  conductance- viscosity-ratio.  Thus  at  o.i  formal  the  co- 
efficient is  less  than  this  ratio  by  6  to  12  percent.  Moreover,  the 
coefficient,  unlike  the  ratio,  passes  through  a  minimum  at  about  0.5 
formal  (except  in  the  case  of  sodium  chloride  where  the  minimum 
occurs  between  i  and  3  formal).  This  comparison  shows  quantita- 
tively the  error  in  assuming  that  the  conductance-viscosity-ratio  is  a 
measure  of  the  ion-activity. 

The  activity-coefficient,  however,  does  not  show,  any  more  than  the 
conductance-ratio,  the  ionization  of  the  substance.  For  just  as  the 
conductance-ratio  affords  no  means  of  determining  the  separate 
influences  of  ionization  and  ion-mobility,  so  these  activity-coefficients 
do  not  show  to  what  extent  the  change  of  activity  with  increasing 
concentration  is  due  to  decreasing  ionization  and  to  what  extent  it  is 
due  to  an  increasing  deviation  in  the  behavior  of  the  ions  from  that  of 
perfect  solutes.  In  other  words,  though  the  activity-coefficient  is  an 
exact  expression  of  the  mass-action  of  the  ions,  it  shows  nothing  about 
their  concentration  or  about  the  concentration  of  the  unionized  sub- 
stance present  with  them.  In  most  applications  of  the  mass-action 
law  it  suffices  to  know  the  activities  of  the  ions  involved,  a  knowledge 
of  the  actual  ionization  being  superfluous.  In  other  applications 
where  the  concentration  of  the  unionized  substance  must  be  evaluated, 
it  is  necessary  to  estimate  the  ionization.  As  there  is  no  experimental 
basis  for  making  this  estimate,  some  supplementary  hypothesis  must 
be  employed  in  these  cases,  for  example,  the  hypothesis  stated  in  Art. 
79  that  the  ionization  of  largely  ionized  substances  is  complete. 

Applications  of  the  Activity  Principles  and  Comparison  of  the  Results 
with  Those  Obtained  by  the  Usual  Methods.  — 

Prob.  58.  —  a.  Calculate  the  activity-coefficient  of  the  OH  ion-con- 
stituent, that  is,  the  ratio  of  its  activity  to  the  activity  or  concentration 
of  the  NH4OH,  in  a  solution  o.i  f.  in  NH4OH  and  0.2  f.  in  NH4C1  at 
25°,  assuming  that  the  NH4+and  Cl~  ions  have  equal  activities  in  the 
solution,  and  taking  i.8Xio~6  as  the  ionization-constant  of  NH4OH. 
b.  Calculate  the  ionization  of  the  NH4OH  in  the  solution  under  the 
assumptions  that  the  conductance-viscosity-ratio  for  NH4C1  is  equal  to 
its  ionization  and  that  its  ions  are  perfect  solutes,  c.  Calculate  the 
ionization  of  the  NH4OH  under  the  assumptions  that  the  NH4C1  is 
completely  ionized  and  that  the  ions  are  perfect  solutes,  d.  Tabulate 
the  results  of  these  three  calculations  so  as  to  show  the  different  conse- 
quences of  the  activity  principle  and  of  the  two  simplifying  assumptions. 


REACTIONS  INVOLVING  SOLIDS  173 

Prob.  59. — When  a  0.06  formal  solution  of  C12  in  water  at  o°  has  come 
to  equilibrium,  it  is  found  (by  the  method  illustrated  in  Prob.  36  of 
Art.  80)  that  31%  of  the  C12  has  undergone  change  according  to  the 
equation  Cl2+H2O=H+Cr  +HOC1.  a.  Calculate  the  equilibrium- 
constant  of  this  reaction  assuming  that  the  HC1  is  completely  ionized 
and  that  its  ions  are  perfect  solutes,  b.  Calculate  under  these  assump- 
tions the  percentage  of  the  C12  that  would  undergo  change  in  a  solution 
that  is  0.06  formal  in  C12  and  0.2  formal  in  HC1  at  the  beginning,  first 
simplifying  the  equation  by  omitting  negligible  quantities,  c.  Calcu- 
late the  equilibrium-constant  using  the  true  activity-coefficient  for  HC1 
obtained  from  the  above  table,  d.  Solve  the  problem  stated  in  6, 
using  true  activity-coefficients,  e.  Tabulate  the  results  of  these  four 
calculations,  so  as  to  show  the  differences  resulting  from  the  use  of  the 
approximate  and  exact  methods  of  treatment. 

Prob.  60.  —  The  solubility  of  T1C1  in  water  is  0.0161  formal  at  25°. 
Calculate  its  solubility  hi  a  0.05  formal  KC1  solution  under  the  following 
assumptions:  a,  that  the  two  salts  are  completely  ionized  and  that  their 
ions  are  perfect  solutes;  b,  that  the  ionization  of  each  salt  is  that  given 
by  the  conductance-viscosity  ratio  for  KC1  at  a  concentration  equal  to 
the  total  salt  concentration  in  the  solution;  c,  that  the  activity-co- 
efficient for  each  salt  is  equal  to  that  for  KC1  at  a  concentration  equal 
to  the  total  salt  concentration  in  the  solution,  d.  Tabulate  the  results 
of  these  three  calculations  together  with  the  experimentally  deter- 
mined solubility,  which  is  0.0059  formal.  (The  deviation  of  the  value 
calculated  with  the  activity-coefficients  from  the  observed  value  is 
probably  due  to  inaccuracy  in  the  assumption  that  the  activity- 
coefficient  for  T1C1  is  equal  to  that  of  KC1  at  the  same  concentration. 


CHAPTER  VIII 

EQUILIBRIUM  OF  CHEMICAL  SYSTEMS  IN  RELATION  TO 
THE  PHASES  PRESENT 


I.    GENERAL  CONSIDERATIONS 

114.  General  Considerations.  —  In  this  chapter  are  considered  the 
principles  relating  to  the  number,  state  of  aggregation,  and  composi- 
tion of  the  phases  (denned  as  in  Art.  35)  which  coexist  in  equilibrium 
with  one  another  when  systems  composed  of  one  substance  or  of  two 
or  more  substances  in  various  proportions  are  subjected  to  different 
temperatures  and  pressures. 

The  kinds  of  phenomena  to  be  considered  are  illustrated  by  the 
following  examples.  The  state  in  which  the  substance  water  exists  is 
determined  by  the  temperature  and  pressure.  Thus  these  conditions 
determine  whether  it  exists  in  the  form  of  a  single  phase  as  ice,  as 
liquid  water,  or  as  vapor;  in  the  form  of  two  phases  as  ice  and  liquid 
water,  as  ice  and  vapor,  or  as  liquid  water  and  vapor;  or  in  the  form 
of  the  three  phases,  ice,  liquid  water,  and  vapor.  So  also  in  the  case 
of  two  substances,  such  as  carbon  bisulfide  and  acetone,  there  are,  as 
shown  by  the  vapor-pressure-composition  and  boiling-point-com- 
position diagrams  of  Arts.  42  and  43,  definite  conditions  of  pressure 
and  temperature  at  which  any  definite  mixture  of  the  two  substances 
forms  two  phases  —  a  liquid  phase  and  a  vapor  phase;  and  under  these 
conditions  the  composition  of  each  phase  is  also  definite. 

A  system  (Art.  23)  is  determined  by  its  composition,  that  is,  by  the 
nature  and  quantities  of  its  components;  the  components  being  any 
substances  out  of  which  the  systems  under  consideration  can  be  made 
up,  through  the  use  of  the  smallest  possible  number  of  such  substances. 
The  state  of  a  system  is  definite  when  it  has  certain  definite  phases 
and  when  the  specific  properties  of  each  phase,  such  as  its  density, 
specific  conductance,  index  of  refraction,  etc.,  have  definite  values. 
In  order  that  the  specific  properties  of  any  one  phase  of  a  system  may 
be  fully  determined,  it  is  necessary  to  specify,  in  addition  to  the  propor- 
tions of  its  components,  any  external  factors  which  affect  these  proper- 
ties. The  only  external  factors  which  commonly  have  an  appreciable 
influence  are  the  pressure  and  temperature;  and,  in  the  following 
considerations  relating  to  the  equilibrium  of  phases,  these  factors 

174 


GENERAL  CONSIDERATIONS  175 

alone  are  taken  into  account,  and  their  values  are  assumed  to  be  uni- 
form throughout  all  the  phases  of  a  system. 

It  is  a  fundamental  law  of  the  equilibrium  between  phases  that  the 
absolute  quantity  of  the  different  phases  does  not  influence  their  com- 
position. Thus  the  composition  of  a  solution  in  equilibrium  with  a 
solid  salt  is  not  dependent  on  the  quantity  of  the  solid  in  contact  with 
the  solution.  The  composition  of  the  vapor  in  equilibrium  with  a 
definite  liquid  solution  is  not  dependent  on  the  quantities  of  liquid 
and  vapor  in  contact  with  one  another.  The  proportion  of  hydrogen 
and  water- vapor  in  equilibrium  with  solid  iron  and  solid  ferrous  oxide 
is  not  dependent  on  the  quantities  of  these  solid  phases  in  contact 
with  the  gaseous  phase.  The  rale  at  which  equilibrium  is  established 
between  phases  is,  however,  greatly  increased  by  increasing  the  extent 
of  the  surfaces  between  them. 

The  purpose  of  this  chapter  may  now  be  more  definitely  stated. 
Its  purpose  is  to  show  how  the  nature,  the  quantity,  and  the  composi- 
tion of  the  phases  of  a  series  of  systems  of  the  same  qualitative  com- 
position vary  when  the  composition  of  the  system  as  a  whole  and  the 
pressure  and  temperature  are  varied.  One-component  systems  are 
first  considered;  then  a  general  principle,  known  as  the  phase  rule, 
applicable  to  systems  with  any  number  of  components,  is  presented; 
and  finally  two-component  and  three-component  systems  are  dis- 
cussed. 


i76 


EQUILIBRIUM  BETWEEN  PHASES 


H.    ONE-COMPONENT   SYSTEMS 

115.  Representation  of  the  Equilibrium-Conditions  by  Diagrams.  — 
In  the  case  of  one-component  systems  the  conditions  under  which 
the  different  phases  exist  in  equilibrium  with  each  other  are  fully 
represented  by  pressure- temperature  diagrams;  for  the  state  of  any 
phase  of  such  systems  is  evidently  determined  when  the  pressure  and 
temperature  are  specified. 

The  figure  shows  a  part  of  the  temperature-pressure  diagram  for 
the  component  sulfur,  which  forms  not  only  liquid  and  gaseous  phases, 
but  also  two  solid  phases,  known  from  their  crystalline  forms  as 
rhombic  and  monoclinic  sulfur.  In  the  diagram  the  (vapor-pressure) 
curve  AB  represents  the  pressures  at  which  rhombic  sulfur  and  sulfur- 
vapor  are  in  equilibrium  at  various  temperatures;  the  (vapor-pressure) 
curve  BC  represents  the  pressures  at  which  monoclinic  sulfur  and 
sulfur- vapor  are  in  equilibrium  at  various  temperatures;  and  the 
(transition-temperature)  curve  BE  represents  the  temperatures  at 
which  monoclinic  and  rhombic  sulfur  are  in  equilibrium  at  various 
pressures.  Temperatures,  like  these,  at  which  two  solid  phases  are 
in  equilibrium  with  each  other  are  called  transition-temperatures. 

0.05- 


0.04- 


0.03- 


0.02- 


0.01- 


Vapor 


90 


95 


115 


120 


125 


100  105  110 

Temperature 

The  point  of  intersection  B  of  these  three  curves  shows  the  only 
temperature  and  pressure  at  which  rhombic  sulfur,  monoclinic  sulfur, 


ONE-COMPONENT  SYSTEMS  177 

and  sulfur-vapor  are  in  equilibrium  with  one  another.    A  point,  like 
this,  at  which  three  phases  coexist  is  called  a  triple  point. 

The  (vapor-pressure)  curve  CD  represents  the  pressures  at  which 
liquid  sulfur  and  sulfur-vapor  are  in  equilibrium  with  each  other  at 
various  temperatures;  and  the  (melting-point)  curve  CF  represents 
the  temperatures  at  which  monoclinic  sulfur  and  liquid  sulfur  are  in 
equilibrium  at  various  pressures.  The  point  C  is  evidently  a  second 
triple  point  at  which  monoclinic  sulfur,  liquid  sulfur,  and  sulfur-vapor 
coexist.  As  indicated  on  the  diagram,  the  fields  between  the  different 
lines  show  the  conditions  under  which  sulfur  exists  as  a  single  phase. 

Prob.  i.  —  Behavior  of  Sulfur  Derived  from  the  Diagram.  —  Describe 
with  the  aid  of  the  diagram  the  changes  that  take  place,  a,  when  sulfur 
is  heated  in  an  evacuated  tube  in  contact  with  its  vapor  from  90°  to 
125°;  b,  when  sulfur  is  allowed  to  cool  from  125°  to  90°,  the  pressure 
being  kept  constant  at  0.04  mm. 

The  curves  BE  and  CF  in  the  figure  are  very  nearly  vertical  lines; 
for  increase  of  pressure  always  produces  a  relatively  small  change 
in  the  transition  or  melting  temperature.  Thus  the  transition-tem- 
perature of  the  two  solid  forms  of  sulfur  increases  about  0.04°,  and 
the  melting-point  of  monoclinic  sulfur  increases  about  0.03°,  per 
atmosphere  of  pressure.  With  certain  substances  the  effect  of  pres- 
sure is  to  decrease  the  transition  or  melting  temperature;  thus  the 
melting-point  of  ice  is  lowered  by  0.0076°  by  an  increase  of  pressure 
of  one  atmosphere.  The  direction  and  magnitude  of  this  effect  can 
be  derived  from  thermodynamic  considerations,  as  shown  in  Art.  165. 
Unstable  Forms.  — 

Prob.  2.  —  a.  To  what  equilibria  do  the  curves  BG,  GC,  and  GH  and 
the  point  G  in  the  figure  correspond?  b.  Considered  with  reference  to 
these  equilibria,  to  what  form  of  sulfur  do  the  fields  EBGH,  HGCF, 
and  GBC  correspond?  c.  In  what  sense  are  these  equilibria  unstable? 
Prob.  j.  —  Draw  a  sulfur  diagram  extending  to  pressures  above  the 
triple-point  for  rhombic,  monoclinic,  and  liquid  sulfur,  which  lies  at  151° 
and  i28oatm. 

Prob.  4.  —  It  will  be  noted  that  the  unstable  form  at  any  temperature 
has  the  greater  vapor-pressure.  Prove  that  this  must  be  so  by  showing 
what  would  happen  if  the  two  forms  were  placed  beside  each  other  in 
an  evacuated  apparatus. 

Prob.  5.  —  Prove  by  a  similar  consideration  that  the  unstable  form 
must  also  have  the  greater  solubility  in  any  solvent,  such  as  carbon 
bisulfide.  (Note  that  a  substance  is  commonly  present  in  the  same 
molecular  form  in  the  solutions  produced  by  dissolving  its  different 
solid  forms.) 


178  EQUILIBRIUM  BETWEEN  PHASES 

How  great  the  tendency  is  for  a  substance  to  remain  in  the  same 
form  after  passing  through  a  melting-temperature  or  transition-tem- 
perature and  thus  to  exist  in  an  unstable  form  depends  in  large  measure 
on  the  nature  of  the  substance.  The  following  general  statements  in 
regard  to  it  can,  however,  be  made.  A  crystalline  solid  cannot  as  a 
rule  be  heated  appreciably  above  its  melting-point;  thus,  ice  always 
melts  sharply  at  o°  (under  a  pressure  of  i  atm.).  On  the  other  hand, 
a  liquid  (like  water)  can  ordinarily  be  cooled  to  a  temperature  con- 
siderably below  the  freezing-point  if  agitation  and  intimate  contact 
with  solid  particles,  especially  with  the  stable  solid  phase,  is  avoided. 
Still  more  pronounced  is  the  tendency  of  solid  substances  to  remain 
in  the  same  form  upon  being  heated  or  cooled  through  a  transition- 
temperature;  thus  rhombic  sulfur  can  be  heated  to  its  melting-point 
(110°),  although  this  is  about  15°  higher  than  the  transition- tempera- 
ture (95.5°)  at  which  it  should  go  over  into  monoclinic  sulfur;  and 
monoclinic  sulfur  can  be  cooled  to  room  temperature  without  going 
over  into  the  rhombic  form,  provided  this  be  done  quickly  and  without 
agitation. 

The  rate  at  which  an  unstable  phase  goes  over  into  the  stable  one 
tends  to  increase  with  the  distance  from  the  transition- temperatures; 
but  when  the  substance  is  below  its  transition-temperature  this 
tendency  may  be  more  than  compensated  by  the  greatly  reduced  rate 
of  reaction  which  a  considerable  lowering  of  temperature  always  pro- 
duces; thus  white  phosphorus  is  an  unstable  form,  but  the  rate  at 
which  it  goes  over  into  the  stable  red  form  at  room  temperature  is  so 
small  that  it  may  be  preserved  unchanged  for  years;  similarly,  diamond 
is  an  unstable  form  of  carbon  at  room  temperature,  but  it  does  not 
go  over  into  graphite  or  amorphous  carbon. 

An  effective  means  of  causing  an  unstable  form  to  go  over  into  a 
stable  one  is  to  mix  it  intimately  with  the  stable  form.  The  transition 
may  also  be  accelerated  by  moistening  the  mixture  of  the  two  forms 
with  a  solvent  in  which  they  are  somewhat  soluble.  These  facts 
are  made  use  of  in  the  determination  of  transition-temperatures. 
Thus  the  transition-temperature  of  sulfur  has  been  determined  by 
charging  a  bulb  with  a  mixture  of  rhombic  and  monoclinic  sulfur, 
filling  it  with  carbon  bisulfide  and  oil  of  turpentine,  keeping  it  for  an 
hour  first  at  95°  and  then  at  96°,  and  noting  whether  the  liquid  rose 
or  fell  in  the  capillary  stem  attached  to  the  bulb.  The  volume  was 
found  to  decrease  steadily  at  95°  (owing  to  the  transition  of  the  mono- 


ONE-COMPONENT  SYSTEMS  179 

clinic  into  the  rhombic  form),  and  to  increase  steadily  at  96°  (owing 
to  the  reverse  transition),  showing  that  the  transition-temperature 
lies  between  95  and  96°. 

Prob.  6.  —  Transition  in  Relation  to  Solubility.  —  a.  Suggest  an 
explanation  of  the  catalytic  action  of  the  solvent  in  accelerating  the 
transition  of  the  sulfur,  b.  Outline  a  method  by  which  the  transition- 
temperature  of  sulfur  could  be  determined  by  quantitative  solubility 
measurements. 


i8o  EQUILIBRIUM  BETWEEN  PHASES 

in.    THE  PHASE  RULE 

116.  The  Concept  of  Variance  and  Inductive  Derivation  of  the 
Phase  Rule.  —  There  will  next  be  considered  a  principle,  called  the 
phase  rule,  which  serves  as  a  basis  of  classification  of  different  types  of 
equilibrium  and  enables  the  number  of  phases  that  exist  under  specified 
conditions  to  be  predicted.  There  must,  however,  first  be  presented 
a  concept  involved  in  that  principle  —  that  of  variance. 

Prob.  7.  —  Number  of  Factors  Determining  the  State  of  One-Component 
Systems.  —  Assuming  that  sulfur  is  kept  at  a  specified  pressure  of  0.04 
mm.,  state  at  what  temperatures  it  is  stable,  a,  in  a  single  phase  as 
rhombic  sulfur,  as  monoclinic  sulfur,  and  as  liquid  sulfur;  b,  in  two 
phases,  as  rhombic  and  monoclinic  sulfur,  and  as  monoclinic  and  liquid 
sulfur;  c,  in  the  three  phases,  rhombic  sulfur,  monoclinic  sulfur,  and 
sulfur-vapor. 

It  will  be  noted  that,  in  order  to  determine  the  position  on  the 
diagram  and  therefore  the  state  of  the  system,  the  values  of  two 
determining  factors,  namely,  the  values  of  both  the  pressure  and 
the  temperature,  must  be  specified  when  there  is  only  one  phase;  that 
the  value  of  only  one  of  these  factors,  either  the  temperature  or  pres- 
sure, need  be  specified  when  any  two  phases  coexist;  and  that  no 
condition  can  be  arbitrarily  specified  when  any  three  phases  coexist. 

The  number  of  determining  factors  whose  values  can  and  must 
be  specified  in  order  to  determine  the  state  of  a  system  consisting  of 
definite  phases  and  components  is  called  its  variance  or  number  of 
degrees  of  freedom;  and,  corresponding  to  the  number  of  such  factors, 
systems  are  said  to  be  nonvariant,  tmivariant,  bivariant,  etc. 

It  is  evident  from  the  preceding  statements  that  when  a  one-com- 
ponent system  consists  of  only  one  phase  the  system  is  bivariant^ 
when  it  consists  of  two  phases  it  is  univariant,  and  when  it  consists 
of  three  phases  it  is  nonvariant.  In  other  words,  the  sum  of  the 
variance  and  number  of  phases  is  always  three  for  a  one-component 
system. 

Relation  of  Variance  to  Number  of  Phases.  — 

Prob.  8.  —  Discuss  with  reference  to  the  principle  just  stated  and  to 
the  sulfur  diagram  the  possibility  of  the  coexistence,  a,  of  the  three 
phases,  rhombic,  monoclinic,  and  liquid  sulfur;  b,  of  the  four  phases, 
rhombic,  monoclinic,  liquid,  and  gaseous  sulfur. 

Prob.  p.  —  Consider  the  following  systems  existing  in  the  phases  in- 
dicated: 


THE  PHASE  RULE  181 

(1)  Solution  of  C2H6OH  and  H2O. 

(2)  Two  solutions,  Br2  in  H2O,  and  H20  in  Br2. 

(3)  Two  solutions,  Br2  in  H2O,  and  H2O  in  Br2;  and  their  vapors. 

(4)  Two  solutions,  Br2  in  H2O,  and  H2O  in  Br2;  their  vapors;  ice 

(5)  Solution  of  H2O,  C2H5OH,  and  HC2H3O2. 

(6)  Solution  of  H2O,  C2H6OH,  and  HC2H3O;  and  their  vapors. 

a.  Upon  the  basis  of  the  principles  already  considered  relating  to 
solubility,  vapor-pressure,  and  freezing-point,  name  definitely  certain 
factors  the  specification  of  whose  values  will  fully  determine  all  the 
properties  of  each  of  these  systems,  b.  Make  a  table  showing  the 
number  of  components,  the  number  of  phases,  the  variance,  and  the 
sum  of  the  number  of  phases  and  the  variance,  for  each  of  these  systems. 
Note  that  the  specific  properties  of  any  one  phase,  such  as  its  density, 
specific  heat-capacity,  or  refractive  index,  have  definite  values  only 
when  it  is  at  some  definite  temperature  and  pressure  and  when  it  has  a 
definite  composition,  which  may  be  determined  by  specifying  the  mol- 
fractions  of  all  but  one  of  its  components. 

These  problems  show  that  in  every  case  the  sum  of  the  number  of 
phases  (P)  and  of  the  variance  (V)  i$  greater  by  two  than  the  number 
of  the  components  (C);  that  is,  P-f  V=C+2.  This  principle,  which 
is  called  the  phase  rule,  is  a  general  one,  applicable  to  systems  consist- 
ing of  any  number  of  components  and  of  any  number  of  phases. 

The  phase  rule  furnishes  a  basis  for  the  classification  of  different 
types  of  equilibrium.  It  also  enables  the  number  of  phases  that  can 
exist  under  specified  conditions  to  be  predicted.  The  usefulness  of 
the  phase  rule  itself  is,  however,  often  exaggerated.  Of  primary 
importance  in  the  treatment  of  the  equilibrium  conditions  of  systems 
in  relation  to  the  phases  present  are  the  methods  of  representing  those 
conditions  by  diagrams,  as  described  in  later  articles  of  this  chapter. 

Prob.  10.  —  Number  of  Phases  Permitted  by  the  Phase  Rule.  —  Sodium 
carbonate  and  water  form  solid  phases  of  the  composition  Na2CO3.H2O, 
Na2CO3.7H2O,  and  Na2CO3.ioH2O.  a.  State  how  many  of  these 
hydrates  could  exist  in  equilibrium  with  the  solution  and  ice  under 
a  pressure  of  i  atm.  b.  State  how  many  of  these  hydrates  could  exist 
in  equilibrium  with  water-vapor  at  30°. 

*117.  Discussion  of  the  Concept  of  Components.  —  The  substances 
to  be  regarded  as  the  components,  and  correspondingly  the  number 
of  components,  are  obvious  in  cases,  like  those  already  considered,  in 
which  there  is  no  transformation  of  the  components  into  other  chemical 
substances.  In  cases  where  a  chemical  action  occurs,  the  term  com- 
ponent requires  to  be  more  specifically  defined,  which  may  be  done 


1 82  EQUILIBRIUM  BETWEEN  PHASES 

as  follows.  It  is  evident  from  the  preceding  derivation  of  the  phase 
rule  (and  from  that  in  Art.  118)  that  in  general  any  substances  may  be 
selected  as  the  components,  the  specification  of  the  relative  quantities 
of  which  in  any  phase  suffices  to  determine  fully  the  composition  of 
that  phase;  the  components,  moreover,  being  such  that  this  is  true 
whatever  phase  of  the  system  is  considered.  This  definition  is  illus- 
trated by  the  following  examples. 

The  composition  of  a  gaseous  phase  containing  hydrogen,  oxygen, 
and  water-vapor  in  equilibrium  with  one  another  at  any  definite 
temperature  and  pressure  is  fully  determined  by  specifying  the  relative 
quantities  of  total  hydrogen  and  total  oxygen  present,  either  as  the 
elementary  substances  or  as  constituents  of  the  water;  for  then  the 
proportions  of  the  three  substances  adjust  themselves  so  as  to  satisfy 
the  conditions  of  the  equilibrium  between  them,  for  example,  so  as  to 
satisfy  the  mass-action  law  if  the  substances  are  perfect  gases.  The 
number  of  components  in  this  case  is  therefore  two. 

The  composition  of  a  gaseous  phase  containing  hydrogen,  oxygen, 
and  water-vapor  at  low  temperatures  where  there  is  no  reaction  be- 
tween the  substances  is  evidently  determined  only  when  the  relative 
quantities  of  hydrogen,  oxygen,  and  water  in  the  phase  are  specified. 
The  number  of  components  is  therefore  three  under  these  conditions. 

It  is  obvious  from  these  examples  that  the  number  of  components  is 
determined  not  only  by  the  substances  present,  but  also  by  the  equi- 
libria that  are  established  between  them.  It  is  therefore  advisable 
to  write  chemical  equations  expressing  all  the  equilibria  actually 
established,  so  as  to  make  sure  that  the  proportion  of  every  chemical 
substance  present  in  any  phase  would  be  determined  by  specifying 
that  definite  relative  quantities  of  the  components  adopted  are  present 
in  that  phase,  and  to  make  sure,  on  the  other  hand,  that  more  com- 
ponents have  not  been  adopted  than  are  necessary. 

Another  aspect  of  the  matter  may  be  considered.  In  a  gaseous 
phase  where  hydrogen,  oxygen,  and  water  are  in  chemical  equilibrium, 
one  particular  system  could  be  produced  by  taking  water  as  the  only 
component;  but  the  phase  considerations  of  this  chapter  have  refer- 
ence always  to  a  series  of  systems  of  the  same  qualitative,  but  varying 
quantitative  composition;  and  such  a  series  containing  hydrogen, 
oxygen,  and  water-vapor  in  every  possible  proportion  cannot  be  pro- 
duced out  of  water  alone,  but  can  be  produced  out  of  hydrogen  and 
oxygen.  These  systems  are  therefore  properly  regarded  as  consisting 


THE  PHASE  RULE  183 

of  two  components.  As  another  example  consider  a  gaseous  phase 
containing  HC1,  O2,  C12,  and  H2O.  By  taking  HC1  and  O2  as  two 
components,  the  other  two  substances  can  be  produced  out  of  them 
by  the  reaction  O2+4HC1  =  2C12+2H2O,  but  only  in  equivalent 
proportions.  To  produce  a  system  containing  the  four  substances  in 
any  proportion  whatsoever,  it  is  necessary  to  make  use  of  a  third 
component,  either  C12  or  H2O.  Thus,  either  by  adding  C12  to  a  mixture 
containing  O2  and  HC1  in  any  proportions  and  C12  and  H2O  in  equiva- 
lent proportions,  or  by  removing  C12  from  such  a  mixture,  any  composi- 
tion-whatever can  be  secured.  The  systems  are  therefore  said  to 
consist  of  three  components.  It  may  be  further  noted  that,  though 
the  number  of  components  is  fixed,  there  is  a  certain  arbitrariness  in 
their  selection.  Thus,  although  in  this  example  O2,  HC1,  and  C12 
have  been  used  as  the  components,  any  other  three  of  the  four  sub- 
stances might  be  employed.  It  is,  however,  usual  to  select  as  com- 
ponents the  simplest  substances. 

Determination  of  Components. — 

Prob.  n.  —  Specify  the  components  which  will  produce  phases 
containing  the  following  chemical  substances,  assuming  chemical 
equilibrium  to  be  established  between  them.  In  answering  this  ques- 
tion, write  chemical  equations  expressing  all  the  equilibria  that  may 
be  established  between  these  substances,  making  sure  that  all  the 
substances  can  be  produced  in  any  proportion  from  the  specified  compo- 
nents, and  that  all  those  components  are  necessary,  a.  Gaseous 
H2,  O2,  CO,  CO2,  H2O.  b.  Solution  containing  H2O,  (H2O)2,  NaC1.2H2O, 
NaCl,  Na+,  Cl~.  c.  Solution  containing  H2O,  NH4CN,  NH4+,  CN~, 
NH3,  NH4OH,  HCN. 

Prob.  12.  —  Specify  the  components  of  the  systems  that  exist  in  the 
following  groups  of  phases,  assuming  chemical  equilibrium  to  be  estab- 
lished between  the  substances  named:  a.  Solid  NH4C1,  gaseous  NH3  and 
HC1.  b.  Solid  CaCO3,  solid  CaO,  gaseous  CO2.  c.  Solid  MgSO4-7H2O, 
solid  MgSO4.6H2O,  solution  of  Mg++SO4=  in  water,  water-vapor. 
d.  Solid  carbon;  gaseous  H2O,  H2,  CO,  and  CO2.  e.  Solid  iron,  solid 
FeO,  gaseous  CO  and  CO2.  /.  Solid  iron,  solid  FeO,  solid  carbon, 
gaseous  CO  and  CO2. 

*118.  Derivation  of  the  Phase  Rule  from  the  Perpetual-Motion 
Principle.  —  In  order  to  determine  fully  the  state  of  one-phase  systems 
consisting  of  any  number  C  of  components  (as  defined  in  Art.  117), 
evidently  the  composition  of  the  phase  and  in  addition  any  external 
factors  that  determine  its  properties  must  be  specified.  The  compo- 
sition of  the  phase  is  fully  determined  by  specifying  the  mol-fractions 
of  all  but  one  of  the  components,  that  is,  by  specifying  C  —  i  quanti- 


1 84  EQUILIBRIUM  BETWEEN  PHASES 

ties.  The  only  external  factors  which  commonly  affect  the  properties 
of  a  phase  of  specified  composition  are  temperature  and  pressure;  but 
in  special  cases  certain  other  factors,  some  of  which  are  mentioned 
below,  have  an  appreciable  influence.  Representing  the  number  of 
such  external  determining  factors  by  n,  the  variance  or  total  number 
of  the  independent  variables  that  must  be  specified  is  C—i+n. 
In  any  system,  therefore,  in  which  the  number  of  phases  P  is  one, 
the  sum  of  the  number  of  phases  and  the  variance  V  is  equal  toC  +  n; 
that  is,  P+V  =  C+w;  or,  for  the  common  case  where  pressure  and 
temperature  are  the  only  external  determining  factors,  P+V  =  C  +  2. 
For  example,  the  state  of  one-phase  systems  consisting  of  three  com- 
ponents (i,  2,  3)  is  fully  determined  by  specifying  four  quantities, 
namely,  the  mol-fractions  (x\  and  x%)  of  any  two  of  the  components 
and  the  pressure  (p)  and  temperature  (71);  hence  any  specific  property 
whatever  of  the  system,  such  as  its  density  d,  can  be  expressed  as  some 
function  of  these  four  variables,  such  as  d  =  f(xi,  xz,  p,  T).  In  the 
general  case,  where  the  number  of  components  is  C  and  the  number  of 
external  factors  is  n,  the  function  becomes 

d  =  i(xi}X2,.  .xc-i,p,  T,.  .)• 

The  derivation  of  the  phase-rule  now  consists  in  showing  that  the 
equation  P  -f  V  =  C  +  n,  which  for  one-phase  systems  is  a  result  of  the 
definitions  of  components  and  of  external  factors,  still  holds  true 
whatever  be  the  number  of  phases.  In  order  that  this  may  be  so, 
it  is  evidently  necessary  only  that  each  new  phase  introduced  into  the 
system  shall  dimmish  the  variance  by  one;  for  then  P+V  will  still 
have  its  former  value  C+n.  That  this  is  the  case  can  be  shown  as 
follows. 

Consider  that  a  system  of  C  components  (i,  2,  3,.  .C)  exists  as  a 
gaseous  phase,  and  that  a  new  liquid  or  solid  phase  is  developed  in  it 
(for  example,  by  varying  the  pressure  or  temperature).  Now  the 
partial  pressures  (pi,  p2,.  .pc)  of  the  separate  components  in  the 
gaseous  phase  are,  like  any  other  property  of  the  phase,  functions 
only  of  the  mol-fractions  (xi,  x2).  .xc-i)  and  of  the  external  factors 
(p,  T,.  .).  This  conclusion  may  be  expressed  mathematically  as 
follows: 

pi=ii(xi,  xz,.  .xc-i,p,  T,.  .). 
p2  =  i*(xi,  xz,.  .xc-i,  p,  T,.  .). 


THE  PHASE  RULE  185 

When  the  new,  liquid  or  solid,  phase  is  present  in  equilibrium  with 
the  gaseous  phase,  the  partial  pressure  of  each  component  in  the 
gaseous  phase  is  determined  by  the  mol-fractions  (#/, 00%, .  .x'c-i)  of 
the  liquid  or  solid  phase,  by  the  temperature,  and  by  any  other  external 
determining  factor  which  has  an  appreciable  influence;  for,  if  a  liquid 
or  solid  phase  of  such  composition  as  to  be  in  equilibrium  with  the  gase- 
ous phase  could  also  be  in  equilibrium  at  the  same  temperature  with 
some  other  gaseous  phase  with  different  partial  pressures,  perpetual 
motion  of  the  kind  described  in  Art.  44  could  be  realized.  This  con- 
clusion from  the  perpetual-motion  principle  that  the  partial  pressures 
must  be  fully  determined  by  the  mol-fractions  in  the  liquid  phase  and 
by  the  external  factors  may  be  expressed  mathematically  as  follows: 


pc=fc'(xi',x2f,..x'c-i,p,  r,..). 

By  equating  these  two  sets  of  expressions  for  pi,  p%,.  .pc,  the  follow- 
ing functional  relations  between  the  mol-fractions  in  the  gaseous 
phase  and  those  in  the  liquid  phase  are  obtained: 

fife,  *2,.  .sc-i,  p,  Tr  .)  =  f/fo',  xj,.  .*'c-i,  p,  TV .). 

ft(xi,  x*,.  .«c-i,  p,  T,.  .)  =  fc'(*i',  **',.  .*'c-i,  p,  T,. .). 

fc(*i,  *2,.  .sc-i,  P,  T,.  .)  =  fc'(*i',  a*',.  .*'c-i,  p,  T,.  .). 
That  is,  the  new  phase  gives  us  C  new  functional  equations  in  which 
only  C  - 1  new  variables  (namely,  xi,  x*, .  .  x'c-i)  are  introduced.    By 
combining  these  new  equations  with  one  another  the  new  variables 
may  evidently  be  eliminated,  yielding  a  functional  relation  of  the 

form: 

fi(*i,a*,.  .xc-i,p,  T,.  .)=o. 

This  is  obviously  a  relation  between  the  C—i+n  variables  which 
determine  any  property  of  the  gaseous  phase,  in  the  way  described  in 
the  first  paragraph  of  this  article.  Hence  the  number  of  these  vari- 
ables whose  values  can  and  must  be  specified  to  determine  the  proper- 
ties of  the  gaseous  phase  is  one  less  than  it  was  when  that  phase  was 
alone  present.  And  in  a  similar  way  this  can  be  shown  to  be  true  also 
of  the  properties  of  the  liquid  or  solid  phase.  In  other  words,  the 
new  phase  diminishes  the  variance  of  the  system  by  one,  and  the  sum 
p_j_V  retains  the  value  C  +  n  which  it  had  when  the  system  consisted 


1 86  EQUILIBRIUM  BETWEEN  PHASES 

of  a  single  phase.  And  evidently,  since  each  additional  phase  formed 
within  the  system  will  similarly  decrease  the  variance  by  one,  the 
sum  P+V  will  always  have  the  same  value  C+n. 

In  the  above  derivation  it  was  assumed  that  a  gaseous  phase  was 
present  in  the  system.  It  will  be  noted,  however,  that  the  partial 
pressures  in  the  gaseous  phase  were  employed  only  as  a  means  of  deriv- 
ing functional  relations  between  the  mol-fractions  of  the  components 
in  two  different  phases,  and  that  the  partial  pressures  disappeared 
in  these  relations.  This  indicates,  and  it  can  be  rigorously  shown, 
that  functional  relations  between  the  mol-fractions  of  the  same  form 
as  those  given  above  hold  true  for  any  pair  of  phases;  for  example, 
for  two  liquid  phases.  It  follows,  therefore,  that  the  phase  rule  is 
applicable  to  any  kind  of  system  whatever,  in  the  form  P+V  =  C+w. 

As  has  been  stated,  the  value  of  n  in  the  above  derived  expression 
of  the  phase  rule  is  commonly  2;  for  the  only  external  factors  that 
ordinarily  influence  appreciably  the  state  of  the  system  are  temper- 
ature and  pressure.  In  some  cases,  however,  other  factors  come  into 
play.  For  example,  this  is  sometimes  true  of  intensity  of  illumination 
or  of  electric  or  magnetic  field.  Thus  illumination  of  silver  chloride 
increases  its  dissociation-pressure;  and  under  certain  conditions  an 
electric  discharge  through  an  equilibrium  mixture  of  nitric  oxide, 
nitrogen,  and  oxygen  increases  the  proportion  of  nitric  oxide  in  the 
mixture.  Another  factor  which  makes  the  value  of  n  greater  than  2 
is  introduced  when  different  pressures  are  applied  by  means  of  semi- 
permeable  walls  to  different  phases  of  the  system.  Thus  the  pressure 
of  the  vapor  in  equilibrium  with  a  liquid  is  progressively  increased 
(as  shown  in  Art.  51)  when  the  liquid  is  subjected  to  an  increasing 
pressure  by  means  of  a  piston  permeable  for  the  vapor  only.  A 
common  case  of  this  kind  is  that  where  the  atmosphere  acts  as  such 
a  piston,  exerting  a  pressure  on  the  liquid  and  solid  phases  of  the 
system,  but  not  on  the  components  in  the  vapor  phase. 


TWO-COMPONENT  SYSTEMS  187 

IV.    TWO-COMPONENT  SYSTEMS 

119.  Systems  with  Solid  and  Gaseous  Phases.  —  The  equilibrium 
of  systems  consisting  of  solid  and  gaseous  phases  at  constant  temper- 
ature has  already  been  considered  from  the  mass-action  viewpoint  in 
Art.  no.  Their  equilibrium  in  relation  to  temperature  and  pressure 
will  now  be  considered  from  the  phase-rule  viewpoint. 

When  a  two-component  system  consists  of  three  phases,  the  phase 
rule  evidently  shows  that  the  specification  of  one  of  the  determining 
factors  (for  example,  the  temperature)  fixes  the  state  of  the  system 
and  therefore  the  values  of  the  other  factors  (for  example,  of  the  pres- 
sure and  of  the  composition  of  the  liquid  or  the  gaseous  phase).  The 
pressures  at  which  the  different  groups  of  three  phases  exist  in  equi- 
librium at  various  temperatures  can  therefore  be  represented  by  lines  on 
a  diagram  in  which  the  pressure  and  temperature  are  the  coordinates. 

Dissociation-Pressure  and  its  Relation  to  Temperature.  — 
Prob.  13.  —  Calcium  carbonate  dissociates  at  high  temperatures  into 
CaO  and  CO2.  a.  Show  from  the  phase  rule  that  CaCO8  can  be  heated 
in  CO2  gas  at  a  given  pressure  through  a  certain  range  of  temperature 
without  any  decomposition  taking  place,  b.  Show  also  that  there  is  one 
temperature,  and  only  one,  at  which  a  mixture  of  CaCOs  and  CaO  can 
be  kept  under  CO2  at  the  given  pressure  without  any  change  occurring. 

Prob.  14.  —  The  dissociation-pressure  of  solid  calcium  carbonate 
is         10  180  320  580  760        looo  mm. 

at      600°  800°  840°  880°          896°        910° 

a.  Construct  a  pressure-temperature  diagram  for  this  two-component 
system,  lettering  the  fields  so  as  to  show  what  phases  coexist  at  different 
temperatures  and  pressures,    b.  State  at  what  temperature  the  solid 
begins  to  dissociate  when  it  is  heated  in  air  free  from  carbon  dioxide. 
c.  State  at  what  temperature  it  would  dissociate  completely  when 
heated  in  a  covered  crucible  (so  that  there  is  equalization  of  the  pressure, 
but  no  circulation  of  air  into  the  crucible),    d.  In  lime-burning  state 
what  temperature  would  have  to  be  maintained  in  the  kiln  if  there  were 
no  circulation  of  gases  through  it.    e.  State  what  temperature  would 
have  to  be  maintained  if  the  coal  used  as  fuel  were  burned  to  carbon 
dioxide  with  the  minimum  quantity  of  air  and  the  combustion-products 
were  passed  up  through  the  kiln. 

Prob.  15.—  The  dissociation  pressure  of  silver  oxide  is  o.i  atm.  at  1 16°, 
0.2  atm.  at  132°,  i.o  atm.  at  175°,  and  2.0  atm.  at  197°.  a.  If  finely 
divided  silver  be  heated  in  the  air,  what  proportion  of  it  will  be  finally 
converted  into  oxide  when  the  temperature  is  130°?  when  it  is  140°? 

b.  How  could  silver  oxide  be  heated  to  1 70°  without  any  decomposition? 
Prob.  16.  —  Describe  a  method  by  which  pure  oxygen  can  be  prepared 

from  the  air  with  the  aid  of  the  reaction  2BaO2(s)  =  2BaO(s)+O2(g). 


i88  EQUILIBRIUM  BETWEEN  PHASES 

Prob.  17. —  When  a  precipitate  of  hydrated  manganese  dioxide  is  ig- 
nited in  the  air  it  comes  to  a  constant  weight  corresponding  to  the  com- 
position MnO2  when  the  temperature  is  450°,  and  to  another  constant 
weight  corresponding  to  the  composition  Mn2O3  when  the  temperature 
is  500°;  but  when  ignited  in  oxygen  it  changes  to  MnO2  even  at  500°. 
State  what  these  facts  show  as  to  the  dissociation-pressure  of  MnO2 
into  Mn2O3,  and  that  of  Mn2O3  into  a  lower  oxide. 

Prob.  18.  —  A  method  has  been  suggested  for  the  standardization  of 
sulfuric  acid  solutions  which  consists  in  adding  an  excess  of  ammonia 
solution,  evaporating,  drying  the  residue  at  100°,  and  weighing  it. 
In  drying  the  salt  some  decomposition  according  to  the  reaction 
(NH4)2SO4(s)=NH4HSO4(s)-fNH3(g)  is  likely  to  take  place.  How 
might  the  process  be  modified  so  as  to  hasten  the  drying  and  yet  en- 
tirely prevent  the  decomposition? 

120.  Systems  with  Solid,  Liquid,  and  Gaseous  Phases.  Pressure- 
Temperature  Diagrams.  —  The  two-component  systems  that  involve 
salt-hydrates  are  of  especial  importance.  The  vapor-pressure  rela- 
tions of  these  substances  at  different  temperatures  may  be  repre- 
sented by  pressure-temperature  diagrams,  like  that  shown  on  the 
next  page.  This  diagram  has  reference  to  systems  consisting  of 
the  two  components  disodium  hydrogen  phosphate  (Na2HPO4)  and 
water  (H2O),  for  the  case  that  the  vapor-phase  (V),  which  consists 
only  of  water- vapor,  is  always  present.  These  two  components  form 
the  following  solid  phases:  ice  (I),  anhydrous  salt  (A),  dihydrate 
Na2HPO4.2H2O  (AWj),  heptahydrate  Na2HPO4.7H2O  (AW7),  and 
dodecahydrate  Na2HPO4.i2H2O  (AWja);  also  a  solution-phase  (S)  of 
variable  composition,  approaching  as  one  limit  pure  water,  whose 
vapor-pressure  is  represented  by  the  uppermost  curve  in  the  diagram. 

The  phase  rule  evidently  requires  that  in  systems  of  this  kind,  in 
addition  to  the  vapor-phase,  two  solid  phases,  or  one  solid  and  one 
liquid  phase,  must  be  present  in  order  that  the  pressure  of  the  vapor 
may  have  a  definite  value  at  any  definite  temperature. 

Prob.  IQ.  —  Interpretation  of  the  Vapor-Pressure—Temperature  Dia- 
gram for  Salt-Hydrates.  —  At  30°  one  formula- weight  of  Na2HPO4 
is  placed  in  contact  with  a  large  volume  of  water-vapor  at  i  mm.,  and 
the  volume  of  the  vapor  is  steadily  diminished  (so  slowly  that  equi- 
librium is  established)  until  finally  there  remains  hi  contact  with  the 
vapor  only  the  saturated  solution  (which  contains  the  components  in 
the  ratio  iNa2HPO4 :  33.2H2O).  State,  with  the  aid  of  the  figure,  the 
changes  that  take  place  in  the  pressure  of  the  vapor  and  the  accompany- 
ing changes  that  take  place  in  the  character  of  the  other  phases  in 
contact  with  it,  throughout  the  whole  process. 


TWO-COMPONENT  SYSTEMS 


30 
Temperature 


40 


Prob.  20.  —  Vapor-Pressure-Composition  Diagram  for  Salt-Hy- 
drates. —  a.  Plot  the  pressure  of  the  vapor  (as  ordinates)  against  the 
number  of  formula-weights  of  water  absorbed  by  the  salt  as  abscissas 
for  the  process  described  in  Prob.  19  (up  to  the  point  where  15  formula- 
weights  have  been  absorbed).  Mark  the  lines  on  the  plot  so  as  to  show 
what  phases  are  present  during  each  stage  of  the  process,  b.  Make 
a  similar  plot  for  the  case  that  the  process  described  in  Prob.  19  takes 
place  at  38°  (instead  of  at  30°). 

Prob.  21.  —  Experimental  Determination  of  the  Hydrates  Formed  by  a 
Salt.  —  Describe  an  experimental  method  of  determining  what  hydrates 
of  copper  sulfate  exist  at  25°. 

Prob.  22.  —  Transition-Temperature  of  Salt-Hydrates.  —  Describe 
with  the  aid  of  the  figure  what  changes  take  place  when  a  mixture  of 
heptahydrate  and  dodecahydrate  is  heated  from  30°  to  38°  in  a  sealed 
tube  previously  evacuated. 


I  go  EQUILIBRIUM  BETWEEN  PHASES 

Prob.  23.  —  Conditions  under  which  Salts  are  Efflorescent  or  Hygro- 
scopic.—  At  30°  moist  air  is  in  contact  with  solid  Na2HPO4.7H2O. 
Under  what  conditions  of  humidity,  a,  would  the  salt  remain  unchanged? 

b,  would  it  lose  water?  c,  would  it  absorb  water?     (By  the  humidity  of  a 
gas  is  meant  the  ratio  of  the  pressure  of  the  water- vapor  in  the  gas  to 
the  vapor-pressure  of  water  at  the  same  temperature.) 

Prob.  24.  —  Drying  of  Hydrates  without  Decomposition.  —  a.  State 
the  conditions  under  which  moist  crystals  of  Na2HPO4.i2H2O  could  be 
completely  dried  at  30°  without  danger  of  decomposition,  b.  Describe 
a  method  by  which  these  conditions  could  be  practically  realized. 

Separation  of  Hydrates  from  Solutions.  — 

Prob.  25.  —  State  what  solid  phase  separates  on  evaporating  a  dilute 
solution  of  Na2HPO4,  a,  at  30°,  b,  at  38°. 

Prob.  26.  —  The  equilibrium-pressures  for  the  reaction  CaSO4.2H2O 
(gypsum)  =CaSO4 (anhydrite) +2H2O(g)  and  the  vapor-pressures  of 
pure  water  at  various  temperatures  are  as  follows: 

Temperature  .     .     50°  55°  60°  65° 

Gypsum    ...    80  109  149  204  mm. 

Water    ....    92  118  149  188  mm. 

The  solubility  of  calcium  sulfate  is  so  small  that  the  vapor-pressure  of 
its  saturated  solution  may  be  considered  to  be  identical  with  that  of 
water,  a.  State  what  happens  on  heating  gypsum  from  50°  to  65°  in  a 
sealed  tube  previously  evacuated,  b.  State  what  solid  phase  separates 
when  a  solution  of  calcium  sulfate  is  evaporated  at  55°,  and  at  65°. 

c.  State  what  solid  phase  would  separate  upon  evaporating  the  solution 
at  55°  if,  when  it  became  saturated,  enough  calcium  chloride  were  added 
to  reduce  its  vapor-pressure  by  10%.     Give  the  reasons  in  each  case. 

Prob.  27.  —  Dehydration  of  Organic  Liquids  by  Salts.  —  Determine 
the  extent  to  which  the  mol-percent  of  water  in  isoamyl  alcohol  can  be 
reduced  by  shaking  at  25°  isoamyl  alcohol  saturated  with  water  with  a 
large  excess  of  each  of  the  following  substances:  a,  Na2HPO4.i2H2O; 
bt  Na2HPO4.7H2O;  c,  Na2HPO4.2H2O;  d,  Na2HPO4.  For  the  data  refer 
to  the  diagrams  of  this  article  and  of  Prob.  47,  Art.  47. 

..121.  Systems  with  Solid  and  Liquid  Phases.  Temperature- 
Composition  Diagrams.  —  In  a  two-component  system  the  composi- 
tion of  a  solution  which  coexists  with  its  vapor  and  a  solid  phase  is 
evidently  fully  determined  by  specification  of  the  temperature,  the 
system  being  univariant.  The  relation  between  this  composition 
and  the  temperature  can  therefore  be  represented  by  a  diagram  in 
which  these  factors  are  plotted  as  coordinates. 

It  is  evident  that  the  equilibrium  conditions  of  a  two-component 
system  can  still  be  represented  by  a  temperature-composition  diagram 
when  the  specification  that  the  vapor-phase  is  present  is  replaced  by 


TWO-COMPONENT  SYSTEMS 


191 


the  specification  that  the  pressure  has  some  definite  value  (greater 
than  that  at  which  the  vapor  can  exist).  Moreover,  since  pressure 
has,  as  illustrated  in  Art.  115,  only  a  small  effect  on  equilibria  in 
which  solid  and  liquid  phases  are  alone  involved,  the  lines  on  the 
temperature-composition  diagram  have  substantially  the  same  posi- 
tion when  the  system  is  under  a  pressure  of  one  atmosphere  as  they 
do  when  it  is  under  the  pressure  of  the  vapor.  And  in  practice  com- 
position-temperature diagrams  are  ordinarily  constructed  from  data 
determined  under  the  atmospheric  pressure. 

The  form  of  the  temperature-composition  diagram  varies  greatly 
with  the  character  of  the  solid  phases  which  the  components  are 
capable  of  producing.  The  simplest  type  of  such  a  diagram  is  that 
in  which  the  two  components  A  and  B  do  not  form  any  solid  compound 
with  each  other,  but  separate  from  the  solution  in  the  pure  state. 
This  type  is  illustrated  by  the  figure,  which  shows  the  complete  dia- 
gram for  systems  composed  of  acetic  acid  (A)  and  benzene  (B)  at 
one  atmosphere. 


10       0 
90     100 


Composition 


The  freezing-point  curve  CD  represents  the  composition  of  the 
solutions  (S)  which  are  in  equilibrium  with  solid  A,  and  the  freezing- 
point  curve  DF  represents  the  composition  of  the  solutions  which  are 
in  equilibrium  with  solid  B  at  different  temperatures.  The  point  D  at 


1 92  EQUILIBRIUM  BETWEEN  PHASES 

which  the  solution  is  in  equilibrium  with  the  two  solid  phases  A  and  B 
is  called  the  eutectic  point.  When  a  solution  in  the  condition  corre- 
sponding to  this  point  is  cooled,  it  solidifies  completely  without  change 
of  composition  or  temperature  to  a  mixture  of  the  two  solid  phases  A 
and  B.  This  mixture  is  usually  so  fine-grained  and  intimate  that  it 
differs  markedly  in  texture  from  ordinary  mixtures  of  the  same  solid 
phases.  It  is  called  the  eutectic  mixture,  or  simply  the  eutectic. 

Prob.  28.  —  Eutectics  in  Relation  to  the  Phase  Rule.  —  Show  that  the 
phase  rule  requires  that  a  solution  at  the  eutectic  point  solidify  without 
change  of  temperature  or  composition  when  heat  is  withdrawn  from  it. 

Prob.  2Q.  —  Effect  of  Cooling  a  Solution  Predicted  from  the  Diagram.  — 
A  tube  containing  a  solution  of  75%  of  benzene  and  25%  of  acetic 
acid  at  10°  is  placed  within  an  air  jacket  surrounded  by  a  freezing  mix- 
ture at  — 20°,  so  that  the  system  slowly  loses  heat,  till  its  temperature 
falls  to  — 10°.  Predict  with  the  aid  of  the  figure  the  values  of  the  tem- 
perature and  composition  of  the  solution  at  which  any  phase  appears  or 
disappears.  State  also  the  character  of  the  solid  mixture  finally 
obtained. 

Prob.  jo.  —  Cooling  Curves.  —  a.  On  a  diagram  having  as  ordinates 
the  temperatures  in  degrees  and  as  abscissas  the  time  of  cooling  in 
arbitrary  units  draw  curves  representing  in  a  general  way  the  rate  at 
which  the  temperature  decreases  when  a  solution  of  75%  of  benzene 
and  25%  acetic  acid  is  cooled  as  described  in  Prob.  29,  assuming,  first, 
that  the  liquid  overcools  without  the  separation  of  any  solid  phase;  and 
assuming,  secondly,  that  the  solid  phases  separate  so  that  there  is  always 
equilibrium.  Take  into  account  the  fact  that  on  cooling  a  system 
there  is  always  an  evolution  of  heat  whenever  a  new  phase  separates. 
b.  Draw  on  the  same  diagram,  at  the  right  of  these  curves,  a  new  cool- 
ing curve  showing  how  the  temperature  changes  when  pure  benzene  is 
cooled  from  10°  to  — 10°.  c.  Draw  a  cooling  curve  also  for  the  case 
that  a  solut'on  of  64%  of  benzene  and  36%  of  acetic  acid  is  cooled 
from  10°  to  — 10°. 

The  fields  in  the  diagram  also  become  of  much  significance,  when  the 
composition  represented  by  the  abscissas  is  understood  to  be  that  of  the 
whole  system  —  not  merely  that  of  the  liquid  phase.  Thus,  when  the 
system  consists  of  the  substances  in  such  a  proportion  (thus,  25%  of  B 
to  75%  of  A)  and  is  at  such  a  temperature  (thus,  o°)  that  its  condition 
is  represented  by  a  point  n  within  the  field  CDG,  the  diagram  shows 
that  it  consists  of  the  two  phases,  solid  A  and  solution  S  of  the  compo- 
sition (corresponding  to  the  point  p)  at  which  these  two  phases  are  in 
equilibrium  at  the  given  temperature.  Similarly,  when  the  system 
has  a  composition  and  temperature  corresponding  to  any  point  within 
the  field  GDJH,  it  consists  of  solid  A  and  the  eutectic  mixture  E£, 


TWO-COMPONENT  SYSTEMS  193 

which  always  has  the  composition  corresponding  to  the  point  D.  It 
can  readily  be  shown,  moreover,  that,  when  the  state  of  the  system  is 
represented  by  the  point  n,  the  weight  of  the  solid  A  is  to  the  weight 
of  the  solution  present  as  the  length  of  the  line  joining  n  and  p  is  to  the 
length  of  the  line  joining  m  and  n;  and  similarly,  that  at  any  point 
below  r  on  the  same  ordinate  with  it  the  weight  of  pure  A  is  to  the 
weight  of  the  eutectic  mixture  present  as  the  length  of  the  line  rD  is 
to  that  of  the  line  Gr. 

Prob.  31.  —  Nature  of  the  Solid  Phases.  —  How  do  the  two  rectangular 
fields  at  the  bottom  of  the  figure  differ  with  respect  to,  a,  the  phases 
present;  b,  the  texture  of  the  mixture? 

Prob.  32.  —  Relative  Weights  of  the  Separate  Phases.  —  Show  that 
the  relation  stated  in  the  last  sentence  of  the  preceding  text  is  true. 

Prob.  jj.  —  Eutectic  Composition  and  Temperature  in  Relation  to  the 
Laws  of  Freezing-Point  Lowering. — Lead  (which  melts  at  327°)  and 
silver  (which  melts  at  960°)  form  a  eutectic  which  melts  at  304°.  The 
heat  absorbed  by  the  fusion  of  one  atomic  weight  of  lead  is  1340  cal. 
Calculate  the  composition  of  the  eutectic,  taking  into  account  the  facts 
that  the  first  part  of  a  freezing-point  curve  can  be  located  with  the  aid 
of  the  laws  of  perfect  solutions,  and  that  the  molecules  of  metallic 
elements  in  dilute  metallic  solutions  are  as  a  rule  identical  with  their 
atoms.  Compare  the  calculated  composition  of  the  eutectic  with  that 
(4.7  Ag  to  95.3  Pb)  derived  from  cooling  curves. 

Prob.  34.  —  Metallurgical  Process  of  Enriching  Lead-Silver  Alloys 
Resulting  from  the  Reduction  of  Ores.  —  A  technical  process  of  enriching 
lead-silver  alloys  has  been  based  on  the  facts  stated  in  Prob.  33.  De- 
scribe how  this  process  might  be  applied  to  a  melt  containing  i  at.  wt. 
of  silver  and  99  at.  wts.  of  lead,  and  state  the  extent  to  which  the  per- 
centage of  silver  could  be  thereby  increased. 

The  types  of  temperature-composition  diagrams  to  be  next  con- 
sidered are  those  for  systems  of  which  the  components  form  one  or 
more  compounds  which  separate  as  solid  phases.  The  simplest  case, 
illustrated  by  Prob.  35,  is  that  in  which  the  solid  compounds  melt  on 
heating.  Another  important  case,  illustrated  by  Prob.  36,  is  that  in 
which  a  solid  compound  separates  which,  when  heated,  instead  of 
melting,  undergoes  transition  into  another  solid  compound  and  a 
liquid  phase. 

The  diagrams  for  such  systems  may  be  constructed  either  from 
cooling  curves  or  from  the  results  of  solubility  and  freezing-point 
measurements.  The  former  method  is  used  in  the  case  of  alloys  and 
other  high-melting  systems  (as  in  Prob.  35);  the  latter,  in  the  case  of 
systems  involving  solutions  at  moderate  temperatures  (as  in  Prob.  36). 


1 94 


EQUILIBRIUM  BETWEEN  PHASES 


Prob.  35.  —  Construction  of  Temperature-Composition  Diagrams  from 
Cooling  Curves.  —  The  figure  shows  the  cooling  curves  for  a  series  of 
mixtures  of  magnesium  and  lead  containing  the  atomic  percentages  of 
lead  shown  by  the  numbers  at  the  tops  of  the  curves.  On  a  diagram 
whose  coordinates  are  temperature  and  atomic  percentages  plot  points 
representing  the  temperature  at  which  the  solidification  of  each  mixture 
begins  and  ends.  Draw  in  solid  lines  representing  the  freezing-point 
curves.  Draw  also  dotted  lines  limiting  the  different  fields  (as  was 
done  in  the  benzene-acetic-acid  diagram),  and  letter  the  fields  so  as  to 
show  of  what  the  system  consists  in  each  field. 


700- 


650°- 


600- 
550- 
500' 
450- 
400- 
350° 
300- 
250°- 


25  33.3  40     50 


100 


-700° 
-650° 
-600° 
-550° 
500° 
450° 
-400° 
-350° 
-300° 
250° 


Time 


Construction  of  a  Temperature-Composition  Diagram  for  Salt  Hydrates. 

Prob.  36. — Construct  a  temperature-composition  diagram  for  the  sys- 
tem composed  of  Na2HP04  and  H2O  by  plotting  the  following  values  of 
the  percentage  (100  x)  of  Na2HPO4  in  the  saturated  solution  as  abscissas 
against  the  temperature  (/)  as  ordinates.  The  solid  phase  which  is  in 
equilibrium  with  the  solution  (S)  at  a  pressure  of  i  atm.  is  indicated 
by  the  letters  above  the  data. 


IOOX 

Temp. 

100* 
Temp. 


Ice 

AWu 

AWn 

AWn 

AWn 

I.I 

2.4 

5-5 

IQ.2 

30.0 

0-5° 

0° 

15° 

30° 

35° 

AWi 

AWi 

AWt 

AW  2 

AW  i 

38.0 

43-5 

44-7 

46.6 

48.0 

43° 

48° 

50° 

55° 

60° 

AW-, 

33-7 
38° 


TWO-COMPONENT  SYSTEMS  195 

Draw  in  on  the  diagram  lines  representing  the  equilibrium  conditions, 
and  mark  each  line  with  letters  indicating  the  phases  which  coexist 
under  the  conditions  represented  by  the  line;  and  at  each  triple  point 
designate  the  phases  that  coexist  there,  the  temperature,  and  the 
composition  of  the  solution. 

Prob.  37. — As  a  means  of  determining  what  the  fields  hi  the  dia- 
gram signify  with  respect  to  the  phases  present,  proceed  as  follows. 
a.  State  the  solid  phases  that  separate,  and  the  way  in  which  the  com- 
position of  the  residual  solution  changes,  on  cooling  from  55°  to  —5°  at 
i  atm.  solutions  containing  i%  and  20%  of  Na2HPO4.  Draw  in  on  the 
diagram  lines  bounding  the  fields  within  which  the  phases  ice  I  and 
solution  S  coexist,  within  which  AWi2  and  solution  coexist,  and  within 
which  ice  and  AWi2  coexist;  and  letter  the  fields  correspondingly. 
(Note  that  at  any  definite  temperature  the  composition  limits  of  any 
such  field  are  the  compositions  of  the  two  phases  that  are  in  equilibrium 
at  that  temperature,  since  these  are  the  limiting  compositions  of  mix- 
tures that  can  be  made  up  of  these  two  phases.)  b.  From  a  similar 
consideration  deduce  and  state  the  results  of  cooling  from  55°  to  —5° 
solutions  containing  36%  and  42%  of  Na2HPO4.  Draw  in  on  the 
diagram  lines  bounding  the  fields  within  which  the  phases  AW7  and 
solution  S  coexist,  and  within  which  the  phases  AWi2  and  AW7  coexist. 
(Note  that,  when  the  triple  point  is  reached,  the  phase  AW]2  separates, 
thereby  tending  to  decrease  further  the  percentage  of  Na2HPO4  in  the 
solution;  but,  since  the  composition  must  remain  constant  so  long  as 
three  phases  are  present,  the  phase  AW7  re-dissolves;  and  this  process 
continues  as  heat  is  withdrawn  either  till  the  phase  AW7  is  consumed 
or  till  the  solution  dries  up.)  c.  Finally  state  the  results  of  cooling 
from  55°  to  —5°  a  solution  containing  46%  of  Na2HPO4.  Draw  in  on 
the  diagram  lines  bounding  the  fields  within  which  the  phases  AW2 
and  solution  S  coexist,  and  within  which  the  phases  AW7  and  AW2 
coexist. 

Prob.  38.— Preparation  of  Different  Hydrates  by  Evaporation  at 
Different  Temperatures.  —  Determine  from  the  diagram  within  what 
limits  of  temperature  a  solution  of  Na2HP04  must  be  evaporated  in 
order  to  obtain  in  the  pure  state,  a,  Na2HPO4.i2H2O;  6,  Na2HPO4.7H2O; 
c,  Na?HPO4.2H2O.  (The  conditions  under  which  such  salt  hydrates 
can  be  dried  without  change  in  composition  were  considered  in 
Prob.  24). 

Conditions  of  Unstable  Equilibrium.  — 

Prob.  39. — a.  By  extending  by  dotted  lines  the  appropriate  curves  on 
the  diagram,  determine  the  solubilities,  in  terms  of  the  percentage  of 
anhydrous  salt  in  the  solution,  of  AW7  at  45°  and  50°,  and  of  AW2  at  45° 
and  50°.  b.  Prove  that  in  contact  with  the  solution  AW7  is  unstable 
at  50°  with  respect  to  AW2  by  showing  with  the  aid  of  these  solubility 
values  what  happens  when  AW7  is  placed  in  contact  with  a  solution 
saturated  at  50°  with  respect  to  AW2.  c.  Show  that  at  45°  the  more 
soluble  hydrate  is  again  the  unstable  one  in  contact  with  the  solution. 


196  EQUILIBRIUM  BETWEEN  PHASES 

Prob.  40.  —  By  extending  the  appropriate  curves  on  the  diagram 
predict,  a,  at  what  temperature  salt  would  begin  to  separate  on  cooling 
a  30%  Na2HPO4  solution  in  case  the  stable  hydrate  AWJ2  separates, 
and  in  case  the  solution  remains  supersaturated  with  respect  to  this 
hydrate;  b,  at  what  temperature  the  hydrate  AWi2  would  melt  if  on 
heating  its  transition  into  AW7  and  solution  did  not  take  place. 

Prob.  41.  —  Correlation  of  Vapor-Pressure-Temperature  and  Temper- 
ature-Composition Diagrams.  —  With  the  aid  of  the  diagram  under 
consideration  and  that  of  Art.  120  determine  and  tabulate  for  the 
temperatures  20,  25,  30,  35.5  and  40°  the  vapor-pressures  and  per- 
centage compositions  of  the  saturated  solutions  and  the  nature  of  the 
solid  phases  with  respect  to  which  the  solutions  are  saturated.  (Note 
that  the  effect  of  pressure  on  the  solubility  is  here  neglected.) 

Brief  consideration  may  also  be  given  to  systems  involving  two 
liquid  phases  as  well  as  solid  phases.  Such  systems  are  occasionally 
met  with  among  alloys,  and  frequently  among  non-metallic  substances, 
such  as  bromine  and  water,  or  isoamyl  alcohol  and  water,  which  are 
partially  miscible  with  each  other.  Their  relations  are  illustrated  by 
the  accompanying  diagram  for  the  two  components  lead  and  zinc. 
In  this  diagram  the  curves  CD  and  DE  represent  the  compositions, 
expressed  in  atomic  percentages,  of  the  two  liquid  phases  in  equi- 
librium with  each  other,  and  the  curve  BC  represents  the  composition 
of  solutions  in  equilibrium  with  zinc. 


Pb  100     90      80      70      60      50       40      30      20      10 

Zn    0       10       20      30      40       50       60      70      80      90     100 


200U 


TWO-COMPONENT  SYSTEMS  197 

Prob.  42.  —  Interpretation  of  Diagrams  for  Systems  with  Solid  Phases 
and  Two  Liquid  Phases.  —  a.  With  the  aid  of  the  diagram  state 
all  that  would  happen  on  cooling  from  1000°  to  300°  alloys  containing 
40  and  90  atomic  percents  of  zinc.  b.  State  all  that  would  happen 
on  gradually  adding  zinc  to  molten  lead  at  350°,  at  700°,  and  at  1000°. 
c.  Specify  the  phases  in  which  the  system  exists  when  its  composition 
and  temperature  are  represented  by  any  point  within  each  of  the  fields, 
ABF,  FBJH,  CDE,  BCEG,  and  BGKJ. 

122.  Systems  with  Liquid  and  Gaseous  Phases.  —  Systems  of  two 
components  having  a  gaseous  phase  and  a  liquid  phase  in  which  the 
components  are  miscible  in  all  proportions  have  already  been  con- 
sidered in  Arts.  42  and  43.     The  boiling-point-composition  diagrams 
of  Art.  43,  which  are  temperature-composition  diagrams  at  one  atmos- 
phere, are  further  considered  in  Prob.  43  with  reference  to  the  signifi- 
cance of  the  fields. 

The  vapor-pressure  relations  of  systems  whose  components  have  only 
limited  reusability  in  the  liquid  state  were  briefly  considered  in  Art.  47. 

Prob.  43.  —  Temperature-Composition  Diagrams.  —  Reproduce  in  the 
form  of  separate  sketches  the  three  diagrams  in  the  figure  of  Art.  43, 
and  letter  each  of  the  fields  so  as  to  show  the  phases  in  which  the  system 
exists  when  its  temperature  and  composition  are  represented  by  any 
point  within  the  field. 

123.  Systems  Involving  Solid  Solutions.  —  The  components  some- 
times separate  from  a  liquid  solution  in  the  form  of  solid  solutions, 
instead  of  in  the  form  of  the  pure  solid  substances  or  of  solid  com- 
pounds of  them.     By  solid  solutions  are  meant  physically  homogeneous 
solid  mixtures  of  two  or  more  substances,  that  is,  solid  mixtures  which 
contain  no  larger  aggregates  than  the  molecules  of  the  substances. 

In  their  equilibrium  relations  solid  solutions  closely  resemble  liquid 
solutions;  differing  from  them  mainly  in  the  respect  that  the  equi- 
librium conditions  are  less  readily  established,  owing  to  the  restricted 
molecular  motion  characteristic  of  the  solid  state.  It  is  often  true 
in  solid  systems  that  equilibrium  is  not  established  and  that  the  solid 
phases  fail  to  attain  the  uniform  composition  corresponding  to  equilib- 
rium, unless  the  systems  are  kept  for  a  time  at  a  high  temperature  or  at 
a  temperature  not  far  below  the  melting-point,  where  the  molecules 
still  retain  sufficient  mobility.  The  purpose  of  the  process  of  temper- 
ing or  annealing  solid  substances  is  to  allow  time  for  the  establishment 
of  equilibrium  at  temperatures  so  high  that  it  may  be  rapidly  attained. 

In  solid  solutions  in  which  equilibrium  is  really  attained,  each 


198  EQUILIBRIUM  BETWEEN  PHASES 

component  lowers,  as  in  liquid  solutions,  the  vapor-pressure,  solution- 
tendency,  and  mass-action  or  activity  of  the  other  component.  As  a 
result  of  these  effects  the  freezing-point  of  a  substance  may  be  either 
raised  or  lowered  by  the  addition  of  another  substance,  according  as 
the  solid  solution  which  separates  is  more  or  less  concentrated  than 
the  liquid  solution,  as  is  shown  by  the  following  problem. 

Prob.  44.  —  Raising  of  Freezing-Point  by  Solutes.  —  The  addition 
of  benzene  to  thiophene  raises  the  freezing-point  of  the  thiophene,  owing 
to  the  formation  of  solid  solutions,  a.  Show  how  this  can  be  accounted 
for  by  sketching  a  diagram,  similar  to  that  of  Art.  48,  representing  the 
vapor-pressures  of  pure  thiophene  in  the  liquid  and  solid  states  and  of  a 
liquid  solution  of  benzene  in  thiophene,  and  draw  in  on  the  sketch  a  line 
showing  the  vapor-pressure  of  thiophene  at  various  temperatures  in  a 
solid  solution  of  benzene  and  thiophene  of  the  composition  of  that  which 
separates  from  the  liquid  solution,  b.  Show  that  the  mol-fraction  of 
benzene  must  be  greater  in  the  solid  solution  than  in  the  liquid  solution 
in  order  that  the  freezing-point  may  be  raised,  provided  Raoult's  law 
holds  for  the  solid  solution  as  usually  seems  to  be  approximately  true 
when  the  proportion  of  solute  is  small. 

The  two  components  are  sometimes  soluble  in  each  other  in  the 
solid  state  in  all  proportions,  forming  a  complete  series  of  solid  solu- 
tions. In  most  cases,  however,  each  component  has  only  a  limited 
solubility  in  the  other  solid;  so  that  two  series  of  solid  solutions  result, 
each  covering  only  a  limited  range  of  composition. 

The  temperature-composition  diagrams  representing  the  equi- 
librium conditions  between  solid  solutions  and  liquid  solutions  corre- 
spond completely  with  those  for  liquid  solutions  and  their  vapors, 
since  in  both  cases  there  are  involved  two  phases  of  variable  composi- 
tion. Thus  systems  in  which  there  are  solid  solutions  containing  the 
components  in  all  proportions  may  have  diagrams  of  any  of  the  three 
types  represented  in  the  figure  of  Art.  43 ;  two  of  which  are  illustrated 
by  Probs.  45  and  46.  Systems  in  which  the  components  have  only 
partial  miscibility  in  the  solid  state  are  illustrated  by  Prob.  47,  the 
diagram  of  which  represents  also  certain  other  effects  which  have  been 
already  considered. 

Alloys  with  Complete  Series  of  Solid  Solutions.  — 
Prob.  45.  —  The  cooling  curves  show  that  on  cooling  molten  mixtures 
of  nickel  and  copper  solidification  begins  and  becomes  complete  at  the 
following  temperatures,  a  solid  solution  separating  in  each  case: 
Percentage  of  nickel  o  10  40  70          100 

Solidification  begins         1083°       1140°       1270°       1375°     1452° 
Solidification  ends  1083°       1100°       1185°       1310°     1452° 


TWO-COMPONENT  SYSTEMS 


199 


a.  Draw  on  a  composition-temperature  diagram  two  continuous  curves 
corresponding  to  these  data.  Letter  each  field  so  as  to  show  of  what  the 
system  consists  at  any  point,  b.  State  what  happens  on  slowly  cool- 
ing a  50%  mixture  from  1400°  to  1200°,  giving  the  compositions  of 
the  liquid  and  solid  solutions  in  equilibrium  with  each  other  at  the 
temperatures  at  which  solidification  begins  and  ends,  and  at  1275°. 
(Note  that  the  compositions  of  the  liquid  and  solid  solutions  at  any 
temperature  are  given  by  the  points  corresponding  to  that  temperature 
on  the  upper  and  lower  curves,  since  at  any  temperature  the  composi- 
tion of  the  liquid  phase  is  identical  with  that  of  a  system  from  which 
at  that  temperature  solid  just  begins  to  separate,  and  the  composition 
of  the  solid  phase  is  identical  with  that  of  a  system  which  at  that  tem- 
perature just  completely  solidifies.) 

Prob.  46.  —  Gold  and  copper  form  a  complete  series  of  solid  solutions. 
The  mixture  containing  60  atomic  percent  gold  has  a  constant  melting- 
point  of  880°.  Gold  melts  at  1063°,  and  copper  at  1083°.  Sketch 
the  temperature-composition  diagram. 

Prob.  47.  —  Alloys  Forming  Compounds  and  Limited  Series  of  Solid 
Solutions.  —  In  the  figure,  constituting  the  diagram  for  silver-magne- 
sium alloys,  in  which  the  composition  is  expressed  in  atomic  percentages, 


lOOOn 


900- 


400-p 1 •-! — '"i  ''i r 

Aq  0   10   20   30   40   50   60   70   80   90  100 
Mg  100  90   80   70   60   50   40   30   20   10   0 


400^ 


the  curves  KD,  DL,  and  MF  represent  the  composition  of  solid  solu- 
tions in  equilibrium  with  the  liquid  solutions  whose  composition  is 
represented  by  the  curves  CD,  DE,  and  EF,  respectively,  a.  State 
what  equilibria  are  represented  by  the  curves  AB  and  BC.  b.  State 
what  compounds  are  indicated  by  the  diagram,  c-h.  State  what 


200  EQUILIBRIUM  BETWEEN  PHASES 

happens  on  cooling  slowly  till  complete  solidification  results  a  liquid 
mixture  containing  the  following  atomic  percentages  of  silver:  c,  go; 
d,  70;  e,  55;  /,  50;  g,  25;  h,  20.  i.  Specify  the  phases  in  which  the 
system  exists  when  its  composition  and  temperature  are  represented 
by  any  point  in  each  of  the  fields  CKD,  DLE,  EMF,  QKDLR,  RLMT, 
andBCJH. 


THREE-COMPONENT  SYSTEMS 


201 


V.    THREE-COMPONENT  SYSTEMS 

124.  Systems  with  Liquid  and  Solid  Phases.  Temperature-Com- 
position Diagrams.  —  In  systems  of  three  components  existing  in  one 
solid  phase  and  one  liquid  phase  the  composition  is  not  determined  by 
specification  of  the  pressure  and  temperature,  but  becomes  definite 
when  the  percentage  or  mol-fraction  of  one  of  the  components  is  also 
specified.  With  such  systems  there  must  therefore  be  employed 
diagrams  which  show  how  the  percentages  or  mol-fractions  of  the 
three  components  are  related  to  one  another.  The  most  common 
form  of  diagram,  one  which  has  the  advantage  of  treating  the  three 
components  symmetrically,  is  an  equilateral  triangle,  along  the  three 


90 


80        70 


30        20       10 


60        50       40 

Percent  Sn 

sides  of  which  are  plotted  the  atomic  percentages  or  mol-fractions  of  the 
three  components,  as  in  the  above  figure.  In  such  a  diagram  the 
vertices  then  represent  the  pure  components,  points  on  the  sides 
represent  mixtures  of  each  pair  of  components,  and  points  within  the 
triangle  represent  mixtures  of  all  three  components.  Thus  the  upper 
vertex  would  represent  pure  cadmium  ;  the  point  H,  a  mixture  consist- 
ing of  42%  of  Cd  and  58%  of  Sn;  and  the  point  G,  a  mixture  consisting 
of  22%  of  Cd,  57%  of  Sn,  and  21%  of  Pb. 


202  EQUILIBRIUM  BETWEEN  PHASES 

At  any  one  temperature  the  various  compositions  of  the  liquid 
phase  with  which  any  definite  solid  phase  is  in  equilibrium  are  repre- 
sented on  such  a  diagram  by  a  line.     Thus  the  lines  HJ,  JK,  and  LM 
represent  at  190°  the  compositions  of  the  liquid  with  which  solid 
cadmium,  solid  lead  and  solid  tin,  respectively,  are  in  equilibrium. 
The  point  J  then  evidently  represents  the  composition  of  the  liquid 
with  which  the  two  solid  phases  Cd  and  Pb  are  in  equilibrium  at  190°. 
Similar  lines  can  be  drawn  corresponding  to  various  other  temper- 
atures, thus  giving  a  series  of  isotherms  representing  the  effect  of 
temperature  on  the  equilibrium  of  the  phases.     The  dotted  line  FG 
evidently  shows  the  variation  with  the  temperature  of  the  composition 
of  the  liquid  phase  in  equilibrium  with  the  two  solid  phases  Cd  and 
Pb;  the  line  DG  shows  the  same  thing  for  the  two  solid  phases  Cd 
and  Sn;  and  the  line  EG  for  the  two  solid  phases  Sn  and  Pb.    And 
the  point  G  shows  the  only  temperature  and  liquid  composition  at 
which  the  three  solid  phases  Cd,  Pb,  and  Sn  coexist  in  equilibrium 
with  the  liquid  phase.     The  mixture  of  three  solid  phases  separating 
at  this  point  is  called  the  ternary  eutectic,  and  the  point  itself  the 
ternary  eutectic  point. 

A  more  complete  representation  of  the  effect  of  temperature  is 
secured  by  plotting  temperatures  along  an  axis  perpendicular  to  the 
plane  of  the  composition  triangle.  A  model  in  the  form  of  a  triangular 
prism  thus  results,  whose  horizontal  sections  are  the  isotherms  repre- 
sented in  a  plane  triangular  diagram,  like  that  in  the  figure. 

Prob.  48.  —  Behavior  of  a  Liquid  Ternary  Alloy  on  Cooling  Predicted 
from  its  Temperature-Composition  Diagram.  —  A  completely  liquid 
mixture  of  50  Cd,  30  Sn,  and  20  Pb  is  cooled  till  it  wholly  solidifies. 
a.  State  the  temperature  at  which  solidification  begins  and  the  nature 
of  the  solid  phase  which  then  separates,  b.  State  the  direction  on  the 
diagram  which  on  further  cooling  the  changing  composition  of  the 
liquid  phase  follows.  (Note  that  the  separation  of  the  solid  phase  does 
not  change  the  ratio  of  the  atomic  quantities  of  the  other  two  com- 
ponents in  the  liquid  phase.)  c.  State  the  temperature  at  which  a 
second  solid  phase  begins  to  separate,  the  composition  of  the  liquid 
phase,  and  the  nature  of  the  solid  phase,  d.  Describe  what  happens 
on  further  cooling,  e.  Describe  the  texture  of  the  solid  alloy. 

125.  Systems  with  Gaseous,  Liquid,  and  Solid  Phases  in  Relation 
to  the  Phase  Rule  and  Mass-Action  Law.  —  Three-component  systems 
existing  in  gaseous,  liquid,  and  solid  phases  will  be  here  considered 
only  with  reference  to  the  phase  rule,  and  incidentally  to  the  mass- 
action  law  in  order  to  show  the  relations  between  these  two  principles. 


THREE-COMPONENT  SYSTEMS 


203 


Applications  of  the  Phase  Ride  and  Mass- Action  Law.  — 
Prob.  49.  — a.  Show  by  the  equilibrium  laws  applicable  to  dilute 
solutions  how  the  solubility  of  CaCO3  in  water  saturated  with  CO2 
gas  at  a  given  temperature  is  quantitatively  related  to  the  pressure  of 
the  CO2  gas,  considering  that  the  salt  in  the  solution  is  substantially 
all  Ca++(HC03~)2.  b.  Show  that  the  phase  rule  requires  that  the 
solubility  of  CaCO3  in  water  saturated  with  CO2  gas  is  definitely  deter- 
mined when  the  pressure  and  temperature  are  specified. 

Prob.  50. —  Silver  chloride  forms  with  ammonia  two  solid  compounds 
AgCl.i|NH3  and  AgC1.3NH3,  which  exist  in  contact  with  aqueous 
solutions.  Determine  by  the  phase  rule  the  variance  of  systems  in 
which  the  following  phases  are  present,  and  illustrate  what  this  variance 
signifies  by  specifying  in  each  case  factors  that  would  determine  the 
state  of  the  system:  a,  solid  AgCl,  aqueous  NH3  solution,  vapor; 
b,  solid  AgCl,  solid  AgCl.i|NH3,  solution,  vapor;  c,  solid  AgCl,  solid 
AgCl.i^NHs,  solid  AgC1.3NH3,  solution,  vapor,  d.  At  25°  ammonia 
vapor  is  in  equilibrium  with  the  solids  AgCl  and  AgCl.i|NH3  at  10  mm. 
and  with  the  solids  AgCl.i|NH3  and  AgC1.3NH3  at  105  mm.  State 
under  what  conditions  each  of  these  two  pairs  of  solid  phases  would 
exist  in  equilibrium  with  an  aqueous  solution,  e.  With  the  aid  of  the 
data  of  Prob.  23,  Art.  39,  determine  the  concentration  of  free  NH3  in 
the  aqueous  solution  at  which  AgCl  is  converted  into  AgCl.i|NH3  at 
25°.  /.  Assuming  that  the  AgCl  in  the  solution  is  substantially  all  in 
the  form  of  completely  ionized  Ag(NH3)2+  Cl~,  derive  from  the  equi- 
librium laws  of  dilute  solutions  a  relation  between  the  solubility  s  of 
AgCl  and  the  partial  pressure  p  of  NH3  in  the  vapor,  first,  when  the 
solid  phase  is  AgCl,  and  secondly,  when  it  is  AgCl.i^NH3. 

The  preceding  problems  illustrate  the  characteristic  differences 
between  the  conclusions  to  be  drawn  from  the  phase  rule  and  from  the 
mass-action  law.  The  phase  rule  is  qualitative,  its  application  pre- 
supposes no  special  knowledge  beyond  that  of  the  number  of  compo- 
nents and  phases,  and  it  is  applicable  without  any  limitation  of  con- 
centration. The  mass-action  law  is  quantitative,  its  application 
presupposes  knowledge  of  the  molecular  species  present,  and,  if  nu- 
merical values  are  to  be  computed,  also  of  their  dissociation-con- 
stants; and  it  is  applicable  in  exact  form  only  to  solutions  or  gases 
at  small  concentrations. 


PART  III 

THE  ENERGY  EFFECTS  ATTENDING  CHEMICAL 

CHANGES,   AND    THE   EQUILIBRIUM    OF 

CHEMICAL  CHANGES  IN  RELATION 

TO  THESE   EFFECTS 


CHAPTER  IX 
THE  PRODUCTION  OF  HEAT  BY  CHEMICAL  CHANGES 


I.    THE  FUNDAMENTAL  PRINCIPLES   OF  THERMOCHEMISTRY 

126.  The  First  Law  of  Thermodynamics.  —  The  branch  of  chemistry 
dealing  with  the  heat-effects  and  changes  in  energy-content  attending 
chemical  changes  is  called  thermochemistry.  The  first  section  of  this 
chapter  is  devoted  to  a  consideration  of  the  fundamental  principles 
underlying  the  subject  and  of  the  methods  of  determining  and  express- 
ing thermochemical  quantities.  In  the  second  section  of  the  chapter 
are  presented  some  of  the  more  important  generalizations  derived 
from  the  results  of  thermochemical  measurements. 

The  general  principles  relating  to  energy  which  form  the  basis  of 
thermochemistry  have  already  ^>een  considered  in  Arts.  23  to  25. 
It  will  suffice  to  recall  here  the  following  fundamental  laws  derived 
from  the  law  of  the  conservation  of  energy: 

(1)  The  energy-content  U  of  a  system  js  fully  determined  by  the 
state  of  the  system;  and  correspondingly,  the  change  in  the  energy- 
content  of  a  system  attending  any  change  in  its  state  is  determined 
solely  by  the  initial  and  final  states  of  the  system  irrespective  of  the 
process  by  which  the  change  in  state  takes  place. 

(2)  The  decrease  U\—  Uz  in  the  energy-content  of  a  system  attend- 
ing any  change  in  its  state  is  equal  to  the  sum  of  the  quantities  of  heat 
Q  and  work  W  produced  in  the  surroundings. 

These  principles  constitute  the  first  law  of  thermodynamics.  They 
are  expressed  by  the  following  equation  known  as  the  first-law  equation: 


It  will  be  noted  that,  though  the  change  in  energy-content  is  definite 
for  a  specified  change  in  state,  the  relative  quantities  of  heat  and  work 
produced  may  vary  with  the  process  by  which  the  change  in  state  is 
brought  about.  The  change  in  energy-content  is  therefore  the  more 
fundamental  quantity. 

The  law  of  initial  and  final  states  is  constantly  employed  in  thermo- 
chemistry to  calculate  the  heat-effects  attending  chemical  changes 
that  do  not  take  place  under  conditions  which  make  possible  exact 
calorimetric  measurements.  This  is  illustrated  by  Prob.  i  below  and 
more  fully  by  the  problems  of  Art.  130. 

207 


208   PRODUCTION  OF  HEAT  BY  CHEMICAL  CHANGES 

Prob.  i.  —  Calculation  of  Heat  of  Reactions  with  Aid  of  the  Law  of 
Initial  and  Final  States.  —  When  at  20°  12  g.  of  graphite  are  burned 
with  oxygen  to  form  COa  at  20°  within  a  closed  vessel  so  that  no  work 
is  produced,  94,200  cal.  are  evolved;  and  when  28  g.  of  CO  are  so  burned 
with  oxygen,  68,000  cal.  are  evolved.  Show  by  the  law  of  initial  and 
final  states  how  the  heat  evolved  by  the  burning  of  12  g.  of  graphite  to 
CO  at  20°  within  a  closed  vessel  can  be  calculated. 

The  relations  between  energy-content,  heat,  and  work  were  illus- 
trated in  Prob.  38,  Art.  23,  by  applying  them  to  the  phenomenon  of 
vaporization.  They  are  applied  to  a  chemical  change  in  Prob.  5, 
Art.  128. 

127.  Heat-Effects  Attending  Isothermal  Changes  in  State.  —  The 
heat-effect  attending  a  change  in  the  state  of  a  system  at  a  constant 
temperature  is  ordinarily  experimentally  determined  by  a  combination 
of  the  following  processes.  The  change  in  state  is  first  caused  to  take 
place  within  a  calorimeter  (as  nearly  as  possible)  adiabatically;  and 
the  change  in  temperature  of  the  calorimeter  is  exactly  measured. 
The  quantity  of  heat  which  must  be  imparted  to  or  withdrawn  from 
the  calorimeter  and  its  contents  to  restore  them  to  the  initial  temper- 
ature is  then  accurately  determined,  either  by  direct  measurement 
or  by  calculation  from  the  known  heat-capacities  of  the  substances 
present.  This  quantity  of  heat  is  evidently  equal  to  the  heat-effect 
that  would  attend  the  change  in  state  if  it  took  place  isothermally  at 
the  initial  temperature.  This  method  is  illustrated  in  principle  by 
the  following  problem. 

Prob.  2.  —  Experimental  Determination  of  the  Heat-Effects  Attending 
Isothermal  Changes  in  State.  —  Into  a  calorimeter  containing  5oH2O 
at  20.00°  iKCl  at  20.00°  is  introduced,  and  the  temperature  falls  to 
15.11°.  a.  What  change  in  state  takes  place  in  this  process,  considering 
the  calorimeter  to  be  a  part  of  the  system;  and  what  is  the  heat-effect, 
assuming  that  any  loss  of  heat  by  radiation  has  been  corrected  for? 
b.  What  change  in  state  takes  place  when  iKCl  is  dissolved  in  5oH2O  at 
20°  and  the  resulting  solution  is  brought  to  20°?  c.  In  order  to  cal- 
culate the  heat-effect  attending  this  change  in  state,  what  other 
change  in  state  must  be  combined  with  that  occurring  in  the  cal- 
orimeter? d.  Calculate  the  val.ue  of  this  heat-effect  with  the  aid  of 
such  of  the  following  data  as  may  be  needed:  the  heat-capacity  of  the 
calorimeter  is  19  cal.  per  degree;  the  specific  heat-capacity  of  solid 
potassium  chloride  is  0.166,  of  water  is  i.oo,  and  that  of  the  solution  of 
iKCl  in  50  H2O  -is  0.904  cal.  per  degree. 

Changes  in  state  at  constant  temperature  may  take  place  without 
change  of  volume  or  without  change  of  pressure.  The  different  heat- 


THE  PRINCIPLES  OF  THERMOCHEMISTRY  209 

effects  attending  these  two  different  changes  in  state  are  known  as  the 
heat-effect  at  constant  volume  Qv  and  the  heat-effect  at  constant  pressure  Qp. 

The  heat-effect  at  constant  pressure  is  the  one  which  is  usually 
called  the  heat  of  reaction,  and  the  one  which  is  commonly  recorded 
in  tables  of  constants.  There  is  difference  of  usage  regarding  the 
algebraic  sign  of  the  heat  of  reaction:  when  heat  is  actually  evolved 
by  the  reaction,  the  heat  of  reaction  is  usually  taken  positive  in  thermo- 
chemical  considerations,  but  negative  in  thermodynamic  considera- 
tions. In  this  book  a  uniform  convention,  corresponding  to  the 
thermochemical  one,  is  employed  throughout;  heat-effects  being 
always  considered  positive  when  heat  is  evolved,  as  in  the  combustion 
of  hydrogen  and  oxygen,  and  negative  when  it  is  absorbed  from  the 
surroundings,  as  in  the  vaporization  of  water. 

The  law  that  the  energy-content  of  a  perfect  gas  at  a  definite  tem- 
perature is  independent  of  its  pressure  (Art.  26)  leads  to  the  conclusion 
that  in  the  case  of  reactions  involving  perfect  gases  the  heat-effect 
Qv  at  constant  volume  is  greater  than  the  heat-effect  Qp  at  constant 
pressure  by  an  amount  equal  to  the  work  produced  when  the  reaction 
takes  place  at  constant  pressure.  This  work  can  readily  be  shown 
to  be  equal  to  (Nz—Ni)RT,  where  N2  denotes  the  number  of  mols 
of  gaseous  substances  present  in  the  final  state,  and  Ni  the  number  in 
the  initial  state.  That  is,  Qv  =  Qp+(N2-Ni)RT.  The  derivation  and 
application  of  this  principle  are  illustrated  by  the  following  problems. 

Prob.  3.  —  Relation  between  the  Heats  of  Reaction  of  Gaseous  Substances 
at  Constant  Pressure  and  Constant  Volume.  —  Show  that  RT  is  the 
difference  between  the  quantities  of  heat  evolved  when  at  T  two  mols  of 
CO  and  one  mol  of  O2  at  i  atm.  unite  to  form  CO2  in  one  case  at  constant 
pressure  (for  example,  in  an  open  calorimeter) ;  and  in  another  case  at 
constant  volume  (for  example,  in  a  bomb  calorimeter).  Note  that  the 
change  of  state  which  results  when  the  reaction  takes  place  at  a  constant 
pressure  of  i  atm.  could  also  be  brought  about  by  a  process  consisting 
of  two  steps,  namely,  by  causing  the  mixture  of  CO  and  O?  at  i  atm.  to 
change  at  constant  volume  to  CO2  (whereby  the  pressure  would  become 
f  atm.),  and  then  compressing  it  till  its  pressure  becomes  i  atm. 

Prob.  4.  —  Heats  of  Combustion  at  Constant  Pressure  and  Constant 
Volume.  —  When  i  formula-weight  of  solid  naphthalene  (CioH8)  is 
burnt  with  oxygen  at  20°  in  a  bomb  calorimeter  123,460  cal.  are  evolved. 
Calculate  its  heat  of  combustion  at  constant  pressure. 

128.  Change  in  Energy-Content  and  in  Heat-Content.  —  The 
decrease  in  the  energy-content  attending  any  change  in  the  state  of  a 
system  is  found  simply  by  adding  together  the  heat  evolved  and  the 


210      PRODUCTION  OF  HEAT  BY  CHEMICAt  CHANGES 

work  produced  in  any  process  by  which  that  change  is  brought 
about. 

When  the  change  in  state  takes  place  at  constant  volume,  no  work 
is  involved,  and  therefore  the  heat  Qv  imparted  to  the  surroundings 
is  equal  to  the  decrease  U\—  U2  in  the  energy-content  of  the  system. 
When,  on  the  other  hand,  the  change  takes  place  at  constant  pressure, 
there  is  not  only  a  quantity  of  heat  Qp  imparted  to  the  surroundings, 
but  also  a  quantity  of  work  produced  in  them  equal  to  p(vz  —  vi),  as 
shown  in  Art.  24.  The  decrease  Ui—  U2  in  the  energy-content  of  the 
system  is  then  equal  to  Qp  +  />fe~  ^0- 

Instead  of  determining  and  recording  the  values  of  U\  —  C/2  for 
various  changes  of  state,  it  is  generally  more  convenient  to  employ 
the  values  of  the  quantity  (Ui-\-piVi)  —  (Uz+pzVz).  The  quantity 
U-\-p  v,  like  the  quantity  U,  is  a  property  of  the  system  which  always 
has  a  definite  value  when  the  system  is  in  a  definite  state,  and  which 
always  changes  in  value  by  a  definite  amount  when  the  system  changes 
from  one  state  to  another,  whatever  be  the  process  by  which  the 
change  in  state  is  brought  about;  for  it  is  evident  that  the  pressure  p 
and  the  volume  v,  as  well  as  the  energy-content  U,  have  values  which 
are  determined  by  the  state  of  the  system.  In  other  words,  a  law  of 
initial  and  final  states  applies  to  the  change  in  the  quantity  U+pv, 
just  as  it  does  to  the  change  in  the  quantity  U.  For  brevity,  this 
quantity  U-\-p  v  will  be  represented  by  a  single  letter  H,  and  will  be 
called  the  heat-content  of  the  system;  it  being  understood  that  this 
term  is  a  purely  conventional  one  which  does  not  imply  that  the 
energy  quantity  denoted  by  it  is  a  heat  quantity,  any  more  than  the 
term  energy-content  implies  it. 

The  relation  between  the  decrease  in  heat-content  and  the  decrease 
in  energy-content  attending  any  change  in  state  is  evidently  expressed 
by  the  equation: 

Hi  -H2  =  Ui-  Ut 


Or  writing,  as  will  be  done  throughout  the  remainder  of  this  book, 
—  A#,  —  A£7,  —  A(£z>),  —AN,  etc.,  for  the  decrease  in  the  value  of  any 
quantity  attending  any  change  in  state,  that  is,  for  the  difference 
between  its  value  in  the  initial  state  (denoted  by  subscript  i)  and  its 
value  in  the  final  state  (denoted  by  subscript  2),  this  equation  may  be 
written  in  the  form 


THE  PRINCIPLES  OF  THERMOCHEMISTRY  211 

Substituting  for  —  A£/  the  expression  for  it  given  by  the  first-law 
equation,  there  results 


For  a  change  in  state  for  which  the  initial  pressure  pi  and  the  final 
pressure  pz  have  the  same  value  p  the  decrease  in  heat-content  is 
equal  to  the  heat  evolved  when  the  change  takes  place  at  the  constant 
pressure  p;  for  the  work  W  produced  in  this  process  is  p(v2  —  vi)  and 
hence  equal  to  A  (/>*>)•  That  is,  -A#  =  Q99  where  Q9  is  the  heat  of 
reaction  at  constant  pressure. 

The  fact  that  the  decrease  of  heat-content  is  equal  to  the  commonly 
considered  heat  of  reaction  at  constant  pressure  is  one  of  the  impor- 
tant reasons  for  employing  the  heat-content,  instead  of  the  energy- 
content,  in  thermochemical  and  thermodynamic  considerations. 
Another  advantage  in  using  heat-content  is  that  it  makes  it  unneces- 
sary to  evaluate  the  change  in  volume  attending  the  change  in  state. 
The  employment  of  heat-content  is  preferable  also  because  it  is  related 
more  closely  than  the  energy-content  to  the  concept  of  free  energy 
considered  in  the  next  chapter,  and  because  many  of  the  thermo- 
dynamic relations  assume  a  simpler  form  when  heat-content  and  free 
energy  are  employed. 

The  relations  between  energy-content  or  heat-content  and  the 
quantities  of  heat  and  work  produced  are  illustrated  by  the  following 
problem. 

Prob.  5.  —  Calculation  of  Changes  in  Heat-Content  and  Energy-Content 
from  the  Heat  and  Work  Produced.  —  a.  When  the  reaction  2CO+O2 
=  2C02  takes  placfc  without  change  of  temperature  or  pressure  in  a 
system  consisting  of  2  mols  of  CO  and  i  mol  of  O2  at  20°  and  i  atm., 
136,000  cal.  are  evolved.  What  is  the  decrease  in  the  heat-content, 
and  what  is  the  decrease  in  the  energy-content,  of  the  system? 
b.  When  the  reaction  takes  place  in  the  same  mixture  without  change 
of  temperature  or  volume  135,420  cal.  are  evolved.  What  is  the  decrease 
in  the  heat-content,  and  what  is  the  decrease  in  the  energy-content  in 
this  case?  c.  In  what  respect  does  the  final  state  of  the  system  in  a 
differ  from  that  in  &,  and  what  principle  previously  considered  accounts 
for  the  fact  that  the  changes  in  energy-content  or  in  heat-content  are 
equal  in  the  two  cases. 

129.  Expression  of  Changes  in  Heat-Content  by  Thermochemical 
Equations.  —  In  order  to  express  the  changes  in  heat-content  that 
attend  changes  in  state,  especially  those  involving  chemical  reactions, 
at  any  constant  temperature  and  pressure,  equations  are  conveniently 


212      PRODUCTION  OF  HEAT  BY  CHEMICAL  CHANGES 

employed  in  which  the  heat-contents  of  the  various  substances  involved 
are  represented  by  their  chemical  formulas,  and  in  which  the  change  in 
heat-content  is  shown  by  placing  a  numerical  term  on  the  right-hand 
side  of  the  equation.  For  example,  the  expression 

Fe203(s)-f3CO(g)  =  2Fe(s)+3C02(g)+9ooocal.  (at  20°) 
signifies  that  at  20°  and  i  atm.  (this  pressure  being  understood  unless 
some  other  pressure  is  stated)  the  heat-content  of  one  formula-weight 
of  solid  ferric  oxide  plus  that  of  three  formula- weights  of  gaseous  carbon 
monoxide  is  equal  to  the  heat-content  of  two  formula-weights  of  solid 
iron  plus  that  of  three  formula-weights  of  gaseous  carbon  dioxide  plus 
9000  cal. ;  9000  cal.  being  the  decrease  ( —  A  H)  in  the  heat-content  of  the 
system,  which  is  equal  to  the  heat  evolved  by  the  system  when  the 
reaction  takes  place  at  a  constant  temperature  and  pressure.  Such 
expressions  are  called  thermochemical  equations,  or  specifically,  heat- 
content  equations. 

As  indicated  in  the  preceding  equation,  the  states  of  aggregation  of 
substances  are  shown  by  affixing  to  the  formulas  letters  within  paren- 
theses, as  stated  in  Art.  109.  The  fact  that  a  substance  is  dissolved 
in  x  formula-weights  of  water  is  shown  by  attaching  the  symbol  #Aq 
to  the  formula  of  the  substance.  Thus  the  equation 

KCl(s)+iooAq  =  KCl.iooAq-44oo  cal.  (at  20°) 

signifies  that  when  at  20°  one  formula-weight  of  solid  potassium 
chloride  is  dissolved  in  100  formula-weights  of  water  there  is  an  in- 
crease of  4400  cal.  in  the  heat-content  of  the  system,  corresponding 
to  an  absorption  of  4400  cal.  from  the  surroundings.  When  the  sub- 
stance is  dissolved  in  so  large  a  quantity  of  water  that  the  addition 
of^iore  water  produces  no  appreciable  heat-effect,  the  symbol  °°  Aq 
may  be  attached  to  the  formula  of  the  substance. 

These  thermochemical  equations  can  be  treated  strictly  as  algebraic 
equations,  and  can  be  combined  with  one  another  by  addition  or 
subtraction;  for  every  quantity  in  them  has  a  definite  value  (namely, 
that  of  the  heat-content  of  the  substance  represented  by  the  formula), 
irrespective  of  the  other  quantities  that  occur  with  it  in  the  equations. 

Expression  of  Heat  Data  and  Calculation  of  Heats  of  Reactions  by 
Thermochemical  Equations.  — 

\JProb.  6.  —  The  union  at  20°  and  i  atm.  of  i  g.  of  aluminum  with 
oxygen  is  attended  by  a  heat-evolution  of  7010  cal. ;  and  the  union  of  i  g. 
of  graphite  with  oxygen  to  form  carbon  monoxide  is  attended  by  a  heat- 
evolution  of  2160  cal.  Express  these  data  in  the  form  of  thermo- 


THE  PRINCIPLES  OF  THERMOCHEMISTRY  213 

chemical  equations;  and  calculate  from  them  the  heat  of  the  reaction 
Al203(s)+3C(s)=2Al(s)4-3CO(g)  at  20°. 

Prob.  7.  —  a.  Express  the  following  data  in  the  form  of  thermochemi- 
cal  equations,  employing  the  conventions  described  in  the  preceding  text. 
The  heat  of  formation  of  i  mol  of  gaseous  HC1  from  the  elementary 
substances  is  22,000  cal.  Its  heat  of  solution  in  100  formula-weights  of 
water  is  17,200  cal.  The  heat  of  the  reaction  between  i  mol  of  gaseous 
chlorine  and  a  solution  of  2  formula-weights  of  HI  in  200  formula- 
weights  of  water,  forming  solid  iodine  and  a  dilute  HC1  solution  is 
52,400  cal.  The  heat  of  solution  of  i  mol  of  gaseous  HI  in  100  formula- 
weights  of  water  is  19,200  cal.  b.  By  combining  these  equations 
calculate  the  heat  of  formation  of  i  mol  of  gaseous  HI  from  gaseous 
hydrogen  and  solid  iodine. 

Although  it  is  true  that  only  changes  in  heat-content  can  be  deter- 
mined, yet  it  is  convenient  to  employ  an  arbitrary  scale  of  heat-content 
which  has  as  its  zero-points  the  heat-contents  of  the  various  elementary 
substances  at  the  temperature  under  consideration,  at  a  pressure  of  one 
atmosphere,  and  in  the  form  which  is  most  stable  at  this  temperature 
and  pressure.  Under  this  convention  the  heat-content  (H)  of  any 
compound  substance  is  evidently  equal  to  the  increase  in  heat-content 
(A#)  which  attends  its  formation  out  of  the  elementary  substances; 
and  in  any  thermochemical  equation  the  formula  of  a  substance  may 
evidently  be  replaced  by  the  numerical  value  of  its  heat  of  formation 
with  opposite  sign.  For  example,  the  heat-content  at  20°  of  one 
formula-weight  of  gaseous  hydrogen  bromide,  or  the  numerical  value 
of  the  formula  iHBr(g)  is  —8500  cal.;  for  8500  cal.  is  the  heat  evolved 
when  it  is  formed  out  of  gaseous  hydrogen  and  liquid  bromine  at  20° 
and  i  atmosphere.  Similarly  the  heat-content  of  a  potassium  chloride 
solution  represented  by  the  formula  KCl.iooAq  is  — 101,200  cal.;  f  ,r 
101,200  cal.  is  the  sum  of  the  heat  (105,600  cal.)  evolved  when  one 
formula-weight  of  solid  KC1  is  formed  out  of  solid  potassium  and 
gaseous  chlorine  at  20°  and  of  the  heat  (—4400  cal.)  evolved  when  it  is 
dissolved  in  100  formula-weights  of  water  at  20°.  The  heat  of  for- 
mation of  this  solution  is  expressed  by  the  equation 

K(s)-hja2(g)+iooAq  =  KCLiooAq+ 101,200  cal. 

This  example  shows  that  in  thermochemical  equations  the  symbol 
ffAq  (not  attached  to  another  formula)  has,  like  the  formulas  of 
elementary  substances,  the  value  zero;  for  the  water  represented  by  it 
has  not  been  formed  out  of  its  elements.  Water  which  has  been  so 
formed  is  represented  by  the  formula  xH2O. 


214      PRODUCTION  OF  HEAT  BY  CHEMICAL  CHANGES 

It  is  evident  that  the  employment  of  heats  of  formation  greatly 
simplifies  the  task  of  determining  and  systematizing  thermochemical 
data;  for,  instead  of  measuring  and  recording  the  change  in  heat- 
content  attending  every  chemical  reaction,  it  suffices  to  do  this  for  the 
formation  of  every  compound  out  of  the  corresponding  elementary 
substances.  The  numerical  values  of  the  heats  of  formation  so 
determined,  which  will  be  found  recorded  in  tables  of  physical  and 
chemical  constants,  like  those  of  Landolt-Bornstein,  may  then  be 
substituted  in  any  thermochemical  equation,  and  the  change  in  heat- 
content  attending  the  corresponding  reaction  may  thus  be  calculated. 


gnt  attenam 
JProb.  8.  — 


Calculation  of  Heat  of  Reaction  from  Heats  of  Formation.  — 
Calculate  the  heat-effect  that  attends  at  20°  the  chemical  reaction 
PbS(s)+2PbO(s)=3Pb(s)+SO2(g)  from  the  following  heats  of  forma- 
tion at  20°:  PbO(s),  50,300  cal.;  PbS(s)?  19,300  cal.;  SO2(g),  70,200  cal. 

130.  Indirect  Determination  of  the  Heat-Effects  of  Chemical 
Changes.  —  On  account  of  radiation  errors  the  heat-effect  can  be 
directly  determined  by  calorimetric  measurements  only  for  those 
chemical  changes  which  take  place  completely  within  a  few  minutes, 
and  for  such  changes  only  when  the  temperature  is  not  greatly  different 
from  the  room  temperature.  It  is,  however,  possible  to  calculate  the 
heat-effects  of  many  other  changes  at  or  near  the  room  temperature 
from  those  which  have  been  directly  measured,  by  applying  the  law 
of  initial  and  final  states.  This  has  already  been  exemplified  in  the 
preceding  articles,  and  it  is  more  fully  illustrated  by  the  following 
problems.  The  indirect  method  commonly  employed  for  determining 
heat-effects  at  temperatures  much  higher  or  lower  than  the  room 
temperature  is  described  in  the  next  article. 

Derivation  of  Heats  of  Formation. — 

*Prob.  g.  —  At  20°  the  heat  of  combustion  of  one  mol  of  acetylene 
(C2H2)  is  313,000  cal.  Calculate  its  heat  of  formation.  The  heat  of 
formation  of  iH2O(l)  is  68,400  cal.,  and  that  of  iCO2(g)  from  graphite 
and  oxygen  is  94,200  cal. 

Prob.  10.  —  a.  State  just  what  heats  of  formation  are  denoted  by  the 
second  and  third  terms  in  the  following  equation: 

Zn(s)+2HClooAq=ZnCl2ooAq-f-H2(g)+34,2oo  cal.  (at  20°). 
b.  Write  the  complete  thermochemical  equations  which  express  these 
heats  of  formation,  taking  into  account  the  facts  that  the  heat  of  forma- 
tion of  i  mol  of  gaseous  HC1  is  22,000  cal.  and  that  its  heat  of  solution 
in  a  large  quantity  of  water  is  17,300  cal. 

Prob.  ii.  —  Calculate  the  heat  of  formation  of  iZn(OH)2(s)  from  the 


THE  PRINCIPLES  OF  THERMOCHEMISTRY  215 

following  equation  and  from  the  other  necessary  data,  which  have  been 
given  in  preceding  problems: 

Zn(OH)2(s)  +  2HC1  ooAq  =  ZnCl2  <x> Aq + 2H2O + 19,900  cal. 

Prob.  12.  —  Calculate  the  heat  of  formation  at  20°  of  iH2SO4(l)  from 
the  following  data  at  20°  and  those  given  in  preceding  problems.  The 
heat  of  solution  in  a  large  quantity  of  water  of  iSO2(g)  is  8000  cal.,  and 
that  of  iH2SO4  is  18,000  cal.  One  mol  of  gaseous  C12  acting  on  a  dilute 
solution  of  i  mol  of  SO2  with  formation  of  a  dilute  solution  of  HC1  and 
H2SO4  produces  a  heat-evolution  of  73,900  cal.  (Note  that  when  x 
formula-weights  of  H2O  are  involved  in  the  chemical  reaction  the 
symbol  #H2O  must  appear  in  the  thermochemical  equation,  even  though 
ooAq  may  also  occur  in  it.)  Ans.  192,300  cal. 

Indirect  Determination  of  Heats  of  Reaction.  — 

Prob.  13.  —  A  direct  determination  of  the  heat-effect  of  the  reaction 
CH3CH2OHcx>Aq+O2(g)=CH3C02H«3Aq-f-H2O  is  not  practicable. 
State  what  measurements  could  be  made  which  would  enable  this  heat- 
effect  to  be  calculated;  and  show  how  it  would  be  calculated  from  the 
results  of  such  measurements. 

Prob.  14.  —  Suggest  a  seiies  of  chemical  reactions  from  whose  heat- 
effects,  which  must  be  readily  determinable  in  a  calorimeter,  the  heat 
of  formation  of  iNa2CO3.ioH2O(s)  at  20°  could  be  calculated.  Write 
the  thermochemical  equations,  and  indicate  how  they  would  be  com- 
bined to  yield  the  desired  result. 

131.  Influence  of  Temperature  on  the  Heat-Effects  Attending 
Chemical  Changes.  —  The  heat-effect  at  constant  pressure  of  a 
chemical  change  taking  place  at  any  temperature  can  be  derived  from 
the  heat-effect  at  constant  pressure  at  any  other  temperature  by  the 
following  consideration  of  two  different  processes  resulting  in  the  same 
change  in  state.  In  one  process  cause  the  chemical  change  (for 
example,  the  union  of  i  mol  of  CO  with  ^  mol  of  62)  to  take  place  at 
pressure  p  and  temperature  TI,  and  heat  the  products  (the  carbon 
dioxide)  under  the  pressure  p  to  the  other  temperature  T2;  and,  in  the 
second  process,  heat  the  reacting  substances  (the  carbon  monoxide 
and  oxygen)  under  the  pressure  p  from  T\  to  7*2,  and  cause  them  to 
combine  (forming  carbon  dioxide)  at  the  pressure  p  and  the  temper- 
ature Tz.  Since  in  each  of  these  processes  the  system  changes  from 
the  same  initial  state  (i  mol  of  CO  and  \  mol  of  O2  at  p  and  7\)  to  the 
same  final  state  (i  mol  of  CO2  at  p  and  r2),  the  total  change  in  heat- 
content  must  be  the  same  in  the  two  processes;  and  therefore  the 
heat  evolved  at  the  temperature  Tz  must  differ  from  the  heat  evolved 
at  the  temperature  7\  by  the  same  amount  as  the  heat  absorbed  in 
heating  the  reacting  substances  differs  from  that  absorbed  in  heating 
the  reaction-products  from  7\  to  T2  at  the  constant  pressure  p. 


216      PRODUCTION  OF  HEAT  BY  CHEMICAL  CHANGES 

Effect  of  Temperature  on  Heat  of  Reaction.  — 

Prob.  15.  —  a.  Represent  on  a  diagram,  in  which  the  ordinates  denote 
the  temperature,  and  the  abscissas  the  heat-content  of  the  system, 
how  the  heat-content  changes  during  each  step  of  the  two  processes 
described  in  the  text  for  bringing  about  the  change  in  state,  iCO+|O2 
at  TI  and  p  to  iCO2  at  Tz  and  p.  Make  the  arbitrary  assumption 
that  the  heat-content  of  the  system  in  the  initial  state  is  zero.  b.  With 
the  aid  of  the  diagram  formulate  an  algebraic  relation  between  the 
heats  of  reaction  QTz  and  QTl  at  the  two  temperatures  T2  and  7\  and 
the  heat-capacities  Ci  and  C2  of  the  system  in  its  initial  and  final  states 
respectively,  b.  Noting  that  this  relation  is  exact  only  when  the  heat- 
capacities  do  not  vary  with  the  temperature,  formulate  an  exact 
differential  expression,  and  also  an  integral  which  is  a  general  expression 
'for  the  difference  in  the  heats  of  reaction  at  two  temperatures,  when 
no  change  takes  place  hi  the  state  of  aggregation  between  those  tem- 
peratures. 

Prob.  16.  —  Calculate  the  heat  of  formation  of  iPbO(s)  at  200°  from 
its  heat  of  formation  (50,300  cal.)  at  20°  and  from  the  specific  heat- 
capacities  at  constant  pressure  of  lead  (0.032),  of  oxygen  (0.212),  and 
of  lead  oxide  (0.052).  Ans.  50,000  cal. 

Prob.  17.  —  Calculate  the  heat  of  formation  of  iH20(g)  at  1000°  from 
the  following  data.  The  heat  of  formation  of  iH2O(l)  at  20°  is  68,400 
cal.  Its  heat  of  vaporization  at  100°  is  —  9,670  cal.  The  molal  heat- 
capacity  at  constant  pressure  at  T  is  6.69+0.00077  for  oxygen,  6.54  + 
0.00077  for  hydrogen,  and  8.81  —  0.00197+0.000,002, 227~for  water- 
vapor.  Ans.  59,800  cal. 


THE  RESULTS  OF  THERMOCHEMISTRY  217 

II.     GENERAL   RESULTS   OF   THERMOCHEMICAL   INVESTIGATIONS 

132.  Heat-Effects  Attending  Changes  in  the  State  of  Aggregation 
of  Substances.  —  The  heat  of  vaporization  of  liquid  substances  the 
heat  of  fusion  of  solid  substances,  and  the  heat  of  transition  of  one 
solid  substance  into  another  (as  of  rhombic  into  monoclinic  sulfur) 
are  quantities  which  are  important  in  themselves  and  which  are  fre- 
quently involved  in  calculations  of  the  heat  of  chemical  reactions. 
The  general  statement  can  be  made  in  regard  to  them  that  the  conver- 
sion of  the  form  that  is  stable  at  lower  temperatures  into  that  stable 
at  higher  temperatures  (for  example,  of  ice  into  water,  or  of  rhombic 
into  monoclinic  sulfur)  is  always  attended  by  an  absorption  of  heat. 

The  following  simple  principle,  known  as  Trouton's  rule,  has  been 
discovered  in  regard  to  the  values  of  the  heat  of  vaporization.  The 
ratio  of  the  molal  heat  of  vaporization  of  a  liquid  at  its  boiling-point 
to  its  boiling-point  on  the  absolute  scale  has  approximately  the  same 
value  (namely,  about  —20.5)  for  all  liquids,  except  those  whose 
molecules  are  associated;  that  is,  A&/T  =  approx.  20.5.  The  actual 
values  of  these  quantities  in  the  case  of  five  very  different  liquids 
are  shown  in  the  following  table. 

HEATS  OF  VAPORIZATION  AND  TROUTON'S  RULE 

Substance  AH  T  AH/T 

Bromine  6760  332          20.4 

Benzene  7350          353          20.8 

Carbon  bisulfide  6380          319          20.0 

Ethyl  ether  6260          308          20.3 

Ethyl  formate  7180          327          22.0 

Substances  containing  the  hydroxyl  group,  such  as  water,  alcohols, 
and  acids,  whose  molecules  in  the  liquid  state  are  for  other  reasons 
believed  to  be  associated  (see  Art.  38),  form  marked  exceptions  to 
Trouton's  rule.  Thus  the  value  of  AH/T  is  25.9  for  water,  27.0  for 
ethyl  alcohol,  and  14.9  for  acetic  acid. 

The  molal  heat  of  vaporization  of  a  liquid  or  solid  substance  can  be 
accurately  calculated  by  the  Clapeyron  equation  (Art.  33)  from  the 
change  of  its  vapor-pressure  with  the  temperature. 

The  heat  of  solution  of  substances  is  another  important  quantity. 
In  determining  and  expressing  it  the  quantity  of  solvent  in  which  a 
definite  weight  of  the  substance  is  dissolved  must  be  taken  into  con- 
sideration. The  two  limiting  cases  are  the  heats  of  solution  in  a  very 
large  quantity  of  solvent  and  in  that  quantity  of  solvent  which  forms 


2i8      PRODUCTION  OF  HEAT  BY  CHEMICAL  CHANGES 

with  the  substance  a  saturated  solution.  These  two  heat-effects 
sometimes  have  different  signs.  They  evidently  differ  by  the  heat  of 
dilution  of  the  saturated  solution  with  a  large  quantity  of  water. 

The  dissolving  of  gaseous  substances  in  solvents  is  always  attended 
by  an  evolution  of  heat;  and  the  dissolving  of  solid  substances  in 
solvents  is  usually  attended  by  an  absorption  of  heat.  The  mixing  of 
liquids  may  be  attended  either  by  evolution  or  absorption  of  heat,  as 
explained  in  Art.  42. 

The  heat  of  dilution  of  substances  in  solution  is  also  important; 
for  it  enables  the  heat  of  formation  of  a  solution  of  one  concentration 
to  be  calculated  from  that  of  a  solution  of  another  concentration,  and 
thus  enables  heats  of  reaction  to  be  calculated  at  different  concentra- 
tions. The  heat-effect  attending  the  addition  of  an  equal  volume  of 
water  to  a  concentrated  aqueous  solution  is  often  large;  but  it  becomes 
less  as  the  concentration  diminishes;  and,  after  a  moderately  small 
concentration  (such  as  0.2  jJDrmal)  has  been  attained,  there  is  usually 
only  a  very  small  heat-effect  on  adding  even  a  very  large  quantity  of 
water.  For  example,  on  adding  at  18°  to  i  formula-  weight  of  gaseous 
HC1  or  of  solid  ZnCk  successively  A7V  formula-  weights  of  water, 
there  are  evolved  the  following  quantities  of  heat  (Q)  in  calories: 


5  5          10          30          50      100     200 

Q  for  HC1  .....     14960      1200      .600       360        120       50 
Q  for  ZnCl2  .  .  .  .       7740      1850      1300      2170      1490     820     390 

Prob.  18.  —  Heat  of  Transition  Derived  from  Heats  of  Solution.  —  The 
heat  of  solution  at  20°  in  a  large  quantity  of  chloroform  of  i  at.  wt.  of 
rhombic  sulfur  is  —640  cal.  and  of  i  at.  wt.  of  monoclinic  sulfur  is  —560 
cal.  Show,  by  applying  the  law  of  initial  and  final  states,  what  other 
heat-effect  can  be  derived  from  these  data,  and  what  its  value  is. 

Prob.  ig.  —  Change  of  Heat  oj  Reaction  with  the  Concentration.  — 
The  heat  of  solution  at  18°  of  iZn(s)  in  HC1.2ooAq  is  34,200  cal.  Find 
its  heat  of  solution  in  HCl.sAq,  a,  by  applying  the  law  of  initial  and 
final  states,  and  b,  by  formulating  the  thermochemical  equations 
involved. 

133.  Heats  of  Reaction  in  Aqueous  Solution.  —  The  investigations 
made  of  the  heat-effects  attending  chemical  reactions  in  aqueous 
solution  between  substances  present  at  fairly  small  concentrations 
(0.1-0.3  normal)  have  established  the  following  principles: 

(i)  On  mixing  solutions  of  two  neutral  salts  which  do  not  form  a 
precipitate  by  metathesis  (for  example,  solutions  of  potassium  chloride 
and  sodium  sulfate)  there  is  scarcely  any  heat-effect.  Exceptions  to 


THE  RESULTS  OF  THERMOCHEMISTRY  219 

this  principle  are  met  with  in  the  few  cases  in  which  a  unionized 
salt  is  produced  by  the  metathesis.  Thus  the  metathetical  reaction 
2K+Cl-  +  Hg^(NO3-)2  =  2K+N03-  +  HgCl2  is  attended  by  a  heat- 
evolution  of  12,400  cal. 

(2)  The  heat  of  neutralization  of  a  solution  of  any  largely  ionized 
monobasic  acid  with  a  solution  of  any  largely  ionized  monacidic  base 
(for  example,  of  hydrochloric  acid,  nitric  acid,  etc.,  with  sodium 
hydroxide,  potassium  hydroxide,  etc.)  has  approximately  the  same 
value,  whatever  be  the  acid  or  base.     At  18°  this  nearly  constant  value 
averages  13,800  cal.  per  equivalent  when  the  acid  and  base  solutions 
are  0.12  to.  0.25  normal. 

(3)  When  the  base  or  acid  is  only  partly  ionized  (as  in  the  case  of 
ammonium  hydroxide  or  hydrofluoric  acid),  the  heat-effect  attending 
its  neutralization  with  a  largely  ionized  acid  or  base  is  often  much 
larger  or  smaller  than  that  observed  when  both  acid  and  base  are 
largely  ionized;  thus  the  heat  of  neutralization  of  one  equivalent  of 
ammonium  hydroxide  with  one  of  hydrochloric  acid  is  12,300  cal., 
and  that  of  one  equivalent  of  hydrofluoric  acid  with  one  of  sodium 
hydroxide  is  16,300  cal. 

(4)  When  one  formula-weight  of  a  dibasic  acid  is  neutralized  in 
steps  by  adding  first  one  equivalent  of  a  largely  ionized  base  and  then 
a  second  equivalent,  the  heat-effects  for  the  two  equivalents  of  base 
are  usually  different;  for  example,  at  18°  the  two  heat-effects  are 
14,600  and  16,600  cal.  in  neutralizing  0.28  normal  sulfuric  acid  solution 
with  0.28  normal  sodium  hydroxide  solution,  and  they  are  11,100  and 
9,100  cal.  in  neutralizing  carbonic  acid  with  that  base. 

(5)  When  certain  polybasic  acids  are  neutralized,  there  is  sometimes 
scarcely  any  heat-effect  when  the  second  or  third  equivalent  of  base 
is  added.    Thus  the  successive  heat-effects  when  phosphorous  acid 
(H3PO3)  is  treated  with  sodium  hydroxide  at  18°  are:  14,800  cal.  with 
the  first  equivalent,  13,600  cal.  with  the  second  equivalent,  and  only 
500  cal.  with  the  third  equivalent;  and  with  hypophosphorous  acid 
(H3PO2)  there  is  a  heat-effect  of  15,200  cal.  with  the  first  equivalent 
of  sodium  hydroxide,  and  only  no  cal.  with  the  second  equivalent. 

(6)  With  certain  polybasic  acids  there  is  a  considerable  heat-effect 
when  to  the  solution  of  the  neutral  salt  another  equivalent  of  base  is 
added;  thus,  there  is  a  heat-effect  of  1200  cal.  on  mixing  a  solution 
containing  one  equivalent  of  sodium  hydroxide  with  one  containing 
one  formula- weight  of  sodium  phosphate  (Na3PO4). 


220      PRODUCTION  OF  HEAT  BY  CHEMICAL  CHANGES 

Interpretation  of  the  Heats  of  Reaction  in  Aqueous  Solutions. — 

Prob.  20.  —  a.  Explain  principle  (i)  stated  in  the  preceding  text  with 
the  aid  of  the  ionic  theory,  assuming  that  the  solutions  are  very  dilute. 
b.  What  conclusion  as  to  the  ionization  or  as  to  the  heat  of  ionization 
of  neutral  salts  can  be  drawn  from  the  fact  that  this  principle  holds 
true  even  at  fairly  high  concentrations  (such  as  0.3  normal)?  c.  Write 
a  thermochemical  equation  corresponding  to  the  ionic  reaction  to  which 
the  heat-effect  is  mainly  due  in  the  reaction  cited  as  an  exception  to  the 
principle. 

Prob.  21.  —  Show  that  on  mixing  dilute  solutions  of  two  salts  (such  as 
lead  nitrate  and  potassium  iodide)  which  form  a  precipitate  by  meta- 
thesis there  must  be  a  heat-effect  which  is  substantially  equal,  but 
opposite  in  sign,  to  the  heat  of  solution  of  the  precipitated  substance 
(the  lead  iodide). 

Prob.  22.  —a.  Write  the  ionic  reaction  to  which  the  nearly  constant 
heat  of  neutralization  of  largely  ionized  acids  and  bases  corresponds. 
b.  State  what  other  heat-effect  is  involved  in  the  neutralization  of 
ammonium  hydroxide  (a  slightly  ionized  base)  with  a  largely  ionized 
acid,  and  find  its  value,  c.  What  is  the  heat-effect  that  attends  the 
reaction  between  iNH4Cl  and  iNaOH  in  0.2  normal  solution? 

Prob,  23.  —  Calculate  the  heat  of  ionization  of  iHF  from  the  facts 
that  its  heat  of  neutralization  in  0.28  normal  solution  with  0.28  normal 
NaOH  solution  has  been  found  to  be  16,300  cal,  and  its  ionization  in 
0.28  normal  solution  is  estimated  to  be  5.2%. 

Prob.  24.  —  Calculate  the  heats  of  ionization  at  18°,  a,  of  H2CO3,  and 
b}  of  HCO3~~  from  the  heats  of  neutralization  of  carbonic  acid  given  in 
paragraph  (4).  Carbonic  acid  is  a  very  slightly  ionized  acid;  sodium 
hydrogen  carbonate  solution  is  practically  neutral;  and  the  sodium 
carbonate  in  the  0.07  formal  solution  produced  by  the  neutralization 
is  6.3%  hydrolyzed. 

Prob.  25.  —  Conductance,  transference,  and  reaction-rate  measure- 
ments have  shown  that  sodium  hydrogen  sulfate  in  o.i  formal  solution 
at  1 8°  consists  approximately  of  52%  of  HSOr,  48%  of  SO4=,  and  of 
the  corresponding  amounts  of  Na+  and  H+.  A  calorimetric  measure- 
ment at  1 8°  has  given  the  result  expressed  by  the  equation: 

NaHSO4.6oiAq+NaOH.2ooAq  =Na2SO4.8oiAq+H2O+ 16,620  cal. 

Calculate  the  heat  of  the  reaction  HS04~=H++SO4=,  assuming  that 
the  ionization  of  NaOH,  of  Na2SO4,  and  of  NaHSO4  (into  Na+  and 
HSO4~)  is  complete. 

Prob.  26.  —  State  what  the  observed  heat-effects,  given  in  paragraph 
(5),  attending  the  neutralization  of  phosphorous  acid  and  of  hypo- 
phosphorous  acid  show  as  to  the  existence  of  salts  of  these  acids  in 
solution. 

Prob.  27.  —  Explain  the  fact,  stated  in  paragraph  (6),  that  there 
is  a  large  heat-effect  on  adding  a  solution  of  NaOH  to  one  of 
Na3P04. 


THE  RESULTS  OF  THERMOCHEMISTRY  221 

Prob.  28.  —  Determination  of  Chemical  Equilibria  by  Thermochemical 
Methods.  —  Describe  a  thermochemical  method  of  determining  the 
extent  to  which  acetic  acid  displaces  hydrofluoric  acid  from  sodium 
fluoride  in  dilute  solution.  The  heat  of  neutralization  of  acetic  acid 
with  sodium  hydroxide  at  18°  is  13,230  cal,  and  that  of  hydrofluoric 
acid  is  16,300  caL 

*134.  Applications  of  Thermochemical  Principles.  —  The  following 
problems  illustrate  some  important  applications  of  thermochemical 
principles  to  industrial  chemical  operations.  Thus  the  problems  in- 
clude the  estimation  of  the  maximum  temperature  produced  by 
flames  or  by  the  explosion  of  gaseous  mixtures,  the  determination  of 
the  quantity  of  heat  produced  in  continuous  chemical  processes 
(whose  removal  often  constitutes  one  of  the  main  difficulties  of  large- 
scale  operation),  and  a  consideration  of  the  effectiveness  of  hot  gases 
as  an  evaporating  agent. 

Maximum  Temperature  Producible  by  Flames  and  Explosions.  — 
Prob.  29.  —  Calculate  the  maximum  temperature  that  could  theoreti- 
cally be  attained  in  the  flame  produced  by  burning  at  20°  a  "water-gas  " 
consisting  of  equimolal  quantities  of  hydrogen  and  carbon  monoxide 
with  twice  the  quantity  of  air  required  for  complete  combustion.  As- 
sume that  the  reaction-products  are  not  appreciably  dissociated,  and 
that  air  contains  iO2  to  4N2.  The  molal  heat-capacity  of  C02  at  con- 
stant pressure  at  T  is  7.0+0.00717—  0.000,001, 86  r2.  The  equation 
H2(g)  +  JO2(g)  =  H2O(1)  +  68,400  cal.  holds  true  at  20°.  For  the  other 
data  needed  see  Art.  27  and  previous  problems  of  this  chapter. 
Ans.  i723°A. 

Prob.  30. — a.  Calculate  the  maximum  temperature  and  pressure  that 
could  be  produced  by  the  explosion  within  a  bomb  of  a  mixture  consist- 
ing of  i  mol  of  H2,  %  mol  of  O2,  and  i  mol  of  N2  at  20°  and  100  mm., 
assuming  that  the  water  produced  is  not  appreciably  dissociated. 
b.  The  maximum  pressure  produced  in  an  actual  experiment  was  found 
to  be  840  mm.  Show  how  from  this  result  the  degree  of  dissociation  of 
the  water-vapor  and  the  temperature  of  the  mixture  at  the  moment  of 
the  explosion  can  be  calculated.  (The  equations  should  be  formulated, 
but  they  need  not  be  solved  numerically.) 

Heat  Evolved  by  Continuous  Processes  at  High  Temperatures.  — 
Prob.  31.  —  In  the  Deacon  process  of  making  chlorine,  a  mixture 
of  oxygen  and  hydrogen  chloride  in  the  proportion  £O2 :  iHCl  at  20° 
is  passed  continuously  into  a  vessel  at  386°  containing  a  suitable 
catalyzer.  The  gas  is  passed  so  slowly  that  equilibrium  is  established, 
80%  of  the  hydrogen  chloride  being  converted  into  chlorine  and  water. 
Calculate  the  heat  which  will  be  given  off  from  the  equilibrium  vessel  per 
mol  of  HC1  passed  through,  and  which  must,  to  maintain  a  constant 


222      PRODUCTION  OF  HEAT  BY  CHEMICAL  CHANGES 

temperature,  be  continuously  withdrawn  by  a  suitable  system  of 
cooling.  For  the  data  needed  see  Art.  27  and  previous  problems  of 
this  chapter.  Ans.  1900  cal. 

Prob.  32.  —  In  the  Grillot  contact  process  of  making  fuming  sulfuric 
acid  a  mixture  of  sulfur  dioxide  and  air  passes  through  a  chamber  con- 
taining a  series  of  trays  charged  with  the  platinum  catalyst.  This 
mixture  enters  the  chamber  at  380°  (the  lowest  temperature  at  which  a 
rapid  reaction  will  occur),  and  rises  in  temperature  as  a  result  of  the 
heat  of  reaction.  This  rise  in  temperature  must  not  exceed  100°,  if 
considerable  dissociation  of  the  SO3  is  to  be  avoided.  Calculate  the 
smallest  volume  of  air  that  may  be  present  with  one  volume  of  sulfur 
dioxide  in  the  gas  entering  the  chamber  at  380°  in  order  that  the  rise  of 
temperature  may  not  exceed  100°.  The  heat-effect  of  the  reaction 
SO2(g)+£O2(g)=SO3(g)  is  22,000  cal.  at  380°.  Consider  the  molal 
heat-capacity  of  each  gas  in  the  mixture  to  be  the  same  as  that  of 
nitrogen  or  oxygen.  Assume  that  97%  of  the  SO2  is  .converted  into 
SO3,  and  that  there  is  no  loss  of  heat  by  heat  interchange.  Ans.  29 
volumes. 

Prob.  33.  —  Heat-Effects  in  the  Process  of  Concentrating  Sulfuric  Acid 
by  Hot  Gases.  —  In  the  Gaillard  process  of  concentrating  sulfuric  acid 
a  73%  acid  (corresponding  to  iH2SO4.2H2O)  is  sprayed  at  20°  into  a 
brick-lined  tower,  and  in  falling  through  the  tower  is  caused  to  meet  an 
ascending  current  of  hot  producer-gas  consisting  (mainly)  of  nitrogen 
and  carbon  monoxide,  which  enters  the  bottom  of  the  tower  at  1100° 
and  leaves  the  top  at  200°.  Calculate  how  many  cubic  meters  of 
producer  gas  must  be  supplied  per  kilogram  of  the  73%  acid  to  furnish 
the  heat  necessary  to  concentrate  it  to  91^%  H2SO4  (corresponding  to 
2H2SO4.iH2O),  assuming  the  latter  leaves  the  tower  at  200°,  and  that 
there  is  no  loss  of  heat  by  radiation.  The  heat  evolved  on  adding 
]VH2O(1)  to  iH2SO4(l)  at  20°  has  been  calorimetrically  determined  and 
found  to  be  expressed  by  the  equation  Q  =  17860^7 (^V  +  i. 80).  The 
heat  of  vaporization  of  i  g.  of  water  at  100°  is  —  537  calories.  The  mean 
specific  heat-capacity  of  91^%  H2S04  between  20°  and  200°  is  0.4  cal. 
per  degree.  In  solving  this  problem  write  the  thermochemical  equa- 
tions by  which  iH2SO4.2H2O  at  20°  can  be  converted  into 
and  i|H20(g)  at  200°.  Ans.  3.9. 


CHAPTER  X 

THE  PRODUCTION  OF  WORK   BY   ISOTHERMAL 

CHEMICAL   CHANGES    IN   RELATION  TO  THEIR 

EQUILIBRIUM   CONDITIONS 


I.  THE  SECOND  LAW  OF  THERMODYNAMICS  AND  THE  CONCEPT 

OF  FREE-ENERGY 

135.  The  Second  Law  of  Thermodynamics  and  its  Application  to 
Isothermal  Changes  in  State.  —  This  chapter  is  devoted  to  a  considera- 
tion of  the  production  of  work  by  chemical  changes.  Before  this  can 
be  adequately  considered,  familiarity  with  certain  aspects  of  another 
general  principle  relating  to  energy,  the  so-called  second  law  of  thermo- 
dynamics, is  essential. 

The  first  law  of  thermodynamics  states  that  when  one  form  of 
energy  is  converted  into  another  the  quantity  of  the  form  of  energy 
that  is  produced  is  equivalent  to  the  quantity  of  the  form  that  dis- 
appears; but  it  does  not  indicate  that  there  is  any  other  restriction 
as  to  the  transformability  of  the  different  forms  of  energy.  Expe- 
rience has  shown,  however,  that  while  the  various  forms  of  work  can  be 
completely  transformed  into  one  another  and  into  heat,  thp.  transfor- 
mation of  heat  into  work  is  subject  to  certain  limitations.  Namely, 
it  is  found  that  heat  is  never  transformed  into  work  by  any  cycle  of 
changes  taking  place  in  any  system  kept  in  surroundings  at  any  definite 
temperature;  by  a  cycle  of  changes  being  meant  any  series  of  changes 
in  state  of  such  a  character  that  the  system  finally  returns  to  its  initial 
state. 

If  a  cycle  of  changes  in  state  by  which  heat  is  transformed  into 
work  at  constant  temperature  could  be  realized,  the  cycle  could  be 
indefinitely  repeated  and  produce  work  in  unlimited  quantity,  thus 
constituting  a  form  of  perpetual  motion,  which  may  be  called  perpetual 
motion  of  the  second  kind.  The  impossibility  of  such  perpetual  motion 
is  the  principle  that  was  stated  and  illustrated  in  Art.  44. 

Perpetual  motion  of  the  second  kind,  by  which  work  is  conceived  to 
be  produced  out  of  heat  at  constant  temperature,  is  to  be  distinguished 
from  perpetual  motion  of  the  first  kind  (described  in  Art.  23),  by 
which  work  is  conceived  to  be  produced  without  consuming  energy 
of  any  form. 

Experience  with  processes  taking  place  at  different  temperatures 

223 


224      PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

has  led  to  the  conclusion  that  this  principle  is  a  consequence  of  a  still 
more  general  law,  known  as  the  second  law  of  thermodynamics,  which 
may  be  expressed  as  follows.  A  process  whose  final  result  is  only  a 
transformation  of  a  quantity  of  heat  into  work  is  an  impossibility. 
The  application  of  this  law  to  changes  of  state  taking  place  at  constant 
temperature  will  be  considered  in  this  chapter;  its  application  to 
processes  involving  different  temperatures,  in  chapter  XII. 

Prob.  i.  —  Production  of  Work  from  Heat  at  Constant  Temperature.  — 
When  a  gas  that  is  placed  in  a  reservoir  of  large  heat-capacity  and 
definite  temperature  expands  against  an  external  pressure,  it  produces 
work  and  withdraws  from  the  reservoir  a  quantity  of  heat.  State 
why  this  transformation  of  heat  into  work  is  not  a  contradiction  of  the 
second  law  of  thermodynamics. 

Since  work  is  never  produced  out  of  heat  by  any  isothermal  cycle 
of  changes  in  a  system,  no  change  in  state  at  constant  temperature  can 
produce  more  work  than  must  be  expended  to  restore  the  system  to  its 
initial  state.  In  other  words,  any  isothermal  change  in  state  taking 
place  in  one  direction  (such  as  the  expansion  of  a  gas  from  pi,  vi,  to 
p2)  V2)  at  T)  is  capable  of  producing  a  definite  maximum  quantity  of 
work;  and  to  make  it  take  place  in  the  opposite  direction  (thus,  to 
compress  the  gas  from  pz,  %,  to  pi,  Vi,  at  T),  there  must  be  expended 
a  quantity  of  work  at  least  equal  to  this  maximum.  Whether  this 
maximum  work  is  actually  produced,  or  whether  no  more  than  this 
minimum  work  is  actually  expended,  depends  on  the  way  or  process 
by  which  the  change  in  state  takes  place.  Thus,  the  work  produced 
by  the  change  in  state  of  the  gas  from  pi,  vi,  to  pz,  %,  at  T  depends 
on  the  process  by  which  this  takes  place;  for  example,  upon  the  oppos- 
ing external  pressure,  which  obviously  may  have  any  value  from  zero 
up  to  a  value  only  infinitesimally  less  than  the  pressure  of  the  gas 
itself.  If  the  process  is  such  that  the  maximum  work  is  actually 
produced  or  only  the  minimum  work  is  expended,  it  is  called  a 
reversible  process.  If  the  process  is  such  that  less  than  the  maximum 
work  is  produced  or  more  than  the  minimum  work  is  expended,  it  is 
called  an  irreversible  process.  It  will  be  noted  that  the  criterion  of  the 
reversibility  of  a  change  in  state  is  whether  the  system  can  be  restored 
to  its  initial  state  without  a  net  expenditure  of  work. 

The  following  problems  illustrate  reversible  and  irreversible  pro- 
cesses and  the  external  conditions  which  must  be  fulfilled  in  order 
that  processes  may  be  reversible. 


THE  CONCEPT  OF  FREE  ENERGY  225 

Reversible  and  Irreversible  Processes  Contrasted.  — 

Prob.  2.—N  mols  of  a  perfect  gas  having  a  pressure  of  2  atm.  are 
enclosed  within  a  cylinder  placed  in  a  thermostat  at  a  temperature  T 
and  provided  with  a  weighted  piston.  The  weight  on  the  piston  is 
reduced  so  that  the  piston  exerts  on  the  gas  a  pressure  of  i  atm.,  and 
the  gas  expands  till  its  own  pressure  becomes  i  atm.  Explain  why  this 
process  is  irreversible;  and  state  what  would  have  to  be  true  of  the 
pressure  exerted  by  the  piston  on  the  gas  during  its  expansion  in  order 
that  the  process  might  be  reversible. 

Prob.  3.  —  a.  Derive  an  expression  in  terms  of  N,  R,  and  T  for  the 
work  produced  by  the  irreversible  process  described  in  Prob.  2.  b.  Derive 
a  corresponding  expression  for  the  work  produced  when  the  same  change 
in  state  is  brought  about  by  a  reversible  process,  c.  Find  the  numerical 
ratio  of  these  two  quantities  of  work. 

Prob.  4.  —  Two  Daniell  cells,  each  having  an  electromotive  force  of 
1. 10  volts,  are  connected  in  series  and  are  used  for  charging  a  lead 
storage-cell  having  a  (counter)  electromotive  force  of  2.10  volts.  Ex- 
plain why  the  process  is  not  reversible;  and  state  how  a  number  of 
Daniell  cells  and  a  number  of  storage-cells  could  be  so  arranged  that 
the  latter  might  be  charged  reversibly.  (In  this  case  the  Daniell  cells 
may  be  regarded  as  the  system,  and  the  storage  cells  as  a  part  of  the 
surroundings  in  which  the  electrical  work  is  produced  and  stored.) 

As  illustrated  by  the  preceding  problems,  in  order  that  the  process 
by  which  a  change  in  state  is  brought  about  may  be  reversible,  the 
pressure  externally  applied  must  be  substantially  equal  to  the  pressure 
exerted  by  the  system  itself,  or  the  applied  electromotive  force  must 
be  substantially  equal  to  the  electromotive  force  of  the  cell.  For  the 
change  in  state  will  take  place  in  one  direction  when  the  applied  pres- 
sure or  electromotive  force  is  only  infmitesimally  less  than  that  of  the 
system,  and  in  the  other  direction  when  it  is  only  innnitesimally 
greater,  and  correspondingly,  the  work  produced  by  the  change  in 
state  is  only  infinitesimally  less  than  that  expended  in  restoring  the 
system  to  its  initial  state.  But,  if  the  applied  pressure  or  electro- 
motive force  were  less  by  a  finite  amount  than  that  of  the  system,  the 
quantity  of  mechanical  or  electrical  work  produced  in  the  surround- 
ings would  evidently  not  suffice  to  restore  the  system  to  its  initial 
state;  and  if  in  causing  the  change  to  take  place  hi  the  opposite 
direction,  the  applied  pressure  or  electromotive  force  were  greater 
than  that  of  the  system,  more  work  would  be  withdrawn  from  the 
surroundings  than  the  system  would  be  capable  of  reproducing  on  re- 
verting to  its  original  state. 

It  is  to  be  noted  that  the  term  reversible  is  always  employed,  in 


226      PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

the  sense  above  defined,  to  designate  a  process  of  such  a  character 
that  it  is  possible  to  restore  the  original  condition  of  things  both  in 
the  system  and  its  surroundings.  After  an  irreversible  process  has 
taken  place,  it  is  in  general  possible  to  restore  the  system  to  its  original 
state,  but  only  by  withdrawing  from  the  surroundings  a  larger  quantity 
of  work  than  was  produced  in  them  (and  imparting  to  them  a  corre- 
sponding quantity  of  heat),  so  that  the  original  condition  in  the  sur- 
roundings is  not  reproduced.  When  an  irreversible  change  has  once 
taken  place,  it  is  not  possible  by  any  means  whatever  to  reproduce 
in  their  entirety  the  conditions  that  previously  existed. 

Since  all  actual  processes  are  irreversible,  there  is  constantly  taking 
place  a  decrease  in  the  work-producing  power  of  the  universe.  This 
irreversibility  arises  from  various  causes.  Thus  there  may  be  no 
device  for  producing  the  available  work  (as  when  a  compressed  gas 
escapes  into  the  atmosphere  or  when  coal  burns  in  the  open  air) ;  the 
mechanism  that  is  employed  may  be  imperfect  owing  to  frictional  or 
electrical  resistance;  or  the  pressure  or  electromotive  force  of  the 
system  may  not  be  balanced  by  applying  an  equal  and  opposite 
external  pressure  or  electromotive  force. 

136.  The  Concepts  of  Work-Content  and  Free  Energy.  —  The 
principles  that  a  definite  change  in  state  is  capable  of  producing  a 
definite  quantity  of  work  and  that  this  quantity  of  work  is  realized 
when,  and  only  when,  the  process  by  which  the  change  is  effected  is 
reversible,  are  further  illustrated  by  the  following  problem. 

Prob.  5.  —  Production  of  Definite  Quantity  of  Work  by  Reversible 
Changes  in  State.  —  In  a  voltaic  cell  consisting  of  one  platinum  electrode 
in  contact  with  a  o.i  f.  HC1  solution  arfd  with  hydrogen  gas  at  i  atm. 
and  of  a  second  platinum  electrode  in  contact  with  the  same  HC1  solu- 
tion and  with  hydrogen  gas  at  o.i  atm.,  hydrogen  is  found  to  go  into 
solution  (as  hydrogen-ion)  at  the  first  electrode  and  to  be  evolved  at  the 
second  electrode,  as  a  result  of  the  electromotive  force  which  is  pro- 
duced, a.  Name  the  change  in  state  that  takes  place  in  such  a  cell 
when  one  faraday  of  electricity  passes  through  it  at  18°,  specifying  all 
the  factors  determining  the  change  in  state,  but  disregarding  the 
transference  in  the  solution,  which  in  a  cell  of  this  kind  will  be  shown 
later  to  be  attended  by  no  energy-effect,  b.  Describe  how  the  same 
change  in  state  could  be  brought  about  reversibly  by  a  process  not 
involving  voltaic  action;  and  calculate  the  work  attending  this  process. 
c.  Show  that  perpetual  motion  could  be  realized  if  this  quantity  of 
work  were  not  equal  to  the  work  that  would  be  produced  per  faraday 
of  electricity  when  the  cell  operates  reversibly. 


THE  CONCEPT  OF  FREE  ENERGY  227 

Since  any  definite  change  of  state  in  a  system  is  capable  of  producing 
a  definite  maximum  quantity  of  work  (a  quantity  which  is  never  ex- 
ceeded, whatever  be  the  process  by  which  the  change  in  state  takes 
place),  it  is  to  be  inferred  that  a  system  in  any  definite  state  possesses 
a  certain  power  of  producing  work,  which  may  be  caUed  its  jsork^ 
content^;  and  that  this  changes  by  some  definite  amount  (from 
A^to~At)  when  the  system  undergoes  any  definite  change  in  state, 
whatever  be  the  process  by  which  this  change  takes  place,  and  what- 
ever be  the  quantity  of  work  which  is  actually  produced.  In  other 
words,  the  work-content  of  a  system,  like  its  energy-content  and  its 
heat-content,  is  a  quantity  characteristic  of  the  state  of  the  system; 
and  the  change  in  the  work-content  of  a  system  is  determined  solely 
by  its  initial  and  final  states. 

The  decrease  (Ai-  A2  or  -AA)  in  the  work-content  of  the  system 
attending  a  definite  change  in  its  state  is  evidently  equal  to  themaxi- 
mum  work  which  the  change  is  capable  of  producing,  and  this  is  equal 
to  the  quantity  of  work  WK  which  the  change  actually  produces  when 
it  takes  place  reversibly.  That  is: 

Ai-At  =  -AA  =  W*. 

It  will  be  noted  that,  while  the  First  Law  requires  that  there  be 
a  quantity  of  energy  produced  in  the  surroundings  equal  to  the  de- 
crease of  the  energy-content  of  the  system,  the  Second  Law  does  not 
require  that  there  be  a  quantity  of  work  (W)  produced  equal  to  the 
decrease  in  work-content  (  as  was  illustrated  by  Prob.  3).  The  Second 
Law  requires  only  that  the  quantity  of  work  produced  be  not  greater 
than  the  decrease  in  work-content.  That  is,  W  >  —AA  for  no  process 
whatever;  W=  —AA  for  a  reversible  process;  and  W  <  —AA  for  an 
irreversible  one. 

Just  as  it  is  more  convenient  in  chemical  considerations  to  con- 
sider the  heat-content  rather  than  the  energy-content  of  systems,  so 
there  are  many  advantages  in  considering  in  place  of  the  work-content 
a  quantity  which  differs  from  it,  just  as  the  heat-content  differs  from 
the  energy-content^5y'the  value  of_  i^ 


This  quantity,  which  may  be  called  the  free-energy  content  (F),  01 
simply  the  free-energy  of  the  system,  is  defined  by  the  equation/?  =» 
A_+£y.  Its  value  is  determined  by  the  state  of  the  system,  since  tin 
values  of  A  and  of  pv  are  so  determined;  and  the  decrease  in  its  value 
when  any  change  in  the  state  of  the  system  takes  place  is  evidently 
equal  to  the  work  produced  when  the  change  takes  place  reversibly, 


228      PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

diminished  by  the  increase  of  the  pressure- volume  product;  that  is: 

F1-F2  =  WR-(p2v2-piV1),   or   -AF  =  WR-A(pv). 

The  difference  (Fi—F2)  between  the  free-energy-content  of  a 
system  in  its  initial  state  and  that  in  its  final  state  will  be  called  the 
free-energy  decrease  ( — AF)  attending  the  change  in  state,  irrespective 
of  the  sign  of  its  numerical  value,  which  may  be  either  positive  or 
negative.  The  change  in  state  is  always  considered  to  take  place  at 
some  constant  temperature. 

Prob.  6.  —  Work-Content  and  Free-Energy-Content  in  Relation  to  the 
Work  Produced.  —  a.  What  is  the  decrease  in  joules  in  the  energy- 
content,  heat-content,  work-content,  and  free-energy-content  of  one 
formula-weight  of  water  when  it  changes  from  liquid  water  to  gaseous 
water  at  100°  against  a  constant  pressure  of  i  atm.,  referring  to  Prob. 
38,  Art.  23,  for  the  data  needed?  b.  What  would  be  the  decrease  in 
these  four  quantities  if  the  liquid  water  at  100°  and  i  atm.  changed  to 
gaseous  water  at  100°  and  i  atm.  by  a  process  which  produces  no  work, 
for  example,  by  introducing  the  liquid  water  into  an  evacuated  vessel 
having  a  volume  equal  to  that  of  the  saturated  vapor? 

In  cases  where  different  parts  of  the  system  are  under  different 
pressures,  as  in  the  cell  of  Prob.  5,  the  decrease  in  free  energy  is  de- 
fined to  be  the  quantity  obtained  by  subtracting  from  the  work  pro- 
duced when  the  change  takes  place  reversibly  the  difference  between 
the  sum  of  the  p  v  values  for  all  the  parts  of  the  system  in  its  final 
state  and  the  sum  of  the  p  v  values  for  all  its  parts  in  its  initial  state. 
Therefore,  in  general: 

Fi-F2  =  TFH-(S^2-S/>i^i);   or  -AF  =  WR-A(^pv). 

The  principle  that  the  free-energy  decrease  attending  any  process 
is  determined  solely  by  the  change  in  state  of  the  system  is  the  funda- 
mental principle  on  which  is  based  the  treatment  of  chemical  equi- 
librium and  of  electrochemical  processes  presented  in  this  and  the 
following  chapters.  The  methods  of  determining  the  free-energy 
change  attending  various  kinds  of  changes  in  state  will  be  systemati- 
cally developed,  and  the  applications  of  the  results  to  chemical  and 
electrochemical  problems  will  be  considered. 

A  knowledge  of  the  free-energy  changes  attending  chemical  changes 
is  of  great  importance  for  the  following  reasons.  In  the  first  place,  free 
energy  values  give  directly  the  maximum  quantity  of  work  obtainable 
from  chemical  changes;  for  example,  they  show  how  much  work  can 
be  produced  by  the  combustion  of  coal  or  gasoline  or  from  the  chemical 


THE  CONCEPT  OF  FREE  ENERGY  229 

reactions  taking  place  in  a  storage  cell;  for  the  decrease  in  free-energy 
attending  the  chemical  change,  and  not  its  heat-effect,  determines 
the  theoretical  efficiency  of  a  heat  engine  or  electric  battery.  Secondly, 
free^eriergy  values  enable  the  equilibrium-constants  of  "chemical 
reactions  to  be  calculated;  and  a  knowledge  of  the  equilibrium  condi- 
tions of  chemical  changes  is  fundamental  both  in  its  scientific  aspects 
and  in  its  technical  applications;  thus  the  proper  control  of  a  gas 
producer  or  of  a  blast  furnace  must  be  based  on  a  quantitative  con- 
sideration of  the  equilibria  of  the  chemical  reactions  involved  in  the  V  J 
process.  Thirdly,  free-energy  values  form  the  basis  of  the  science  of 
electrochemistry  on  the  electromotive  force  side;  thus  they  enable 
not  only  the  total  electromotive  force,  but  also  the  separate  potentials, 
of  voltaic  cells  of  all  kinds  to  be  calculated. 


230  PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

II.    FREE-ENERGY    CHANGES    ATTENDING    PHYSICAL    CHANGES 

137.  Free-Energy  Changes  Attending  Changes  in  Volume  and 
Pressure.  —  In  finding  the  expression  for  the  free-energy  decrease 
attending  any  kind  of  change  in  state,  the  method  of  procedure  is 
always  as  follows.  First,  the  change  in  state  of  the  system  is  formu- 
lated by  defining  accurately  its  initial  and  final  states;  second,  a 
process  is  conceived  by  which  the  change  in  state  takes  place  reversibly ; 
and  third,  the  work  attending  this  reversible  process  is  evaluated. 

The  simplest  type  of  change  in  state  is  that  in  which  a  system  whose 
pressure  is  p  and  volume  v  changes  its  state  at  a  constant  temperature 
T  so  that  its  pressure  becomes  p+dp  and  its  volume  v-t-dv.  This 
change  in  state  will  take  place  reversibly  if  a  substantially  equal 
external  pressure  be  applied  to  the  system.  The  work  dWB  in  this 
process  is  equal  to  pdv  (Art.  24).  The  free-energy  decrease  —  dF 
attending  such  a  change  in  state  at  a  constant  temperature  is,  however, 
by  definition  equal  to  dW^—  d(pv).  In  virtue  of  the  mathematical 
relation  d(pv)=pdv-\-vdp,  the  free-energy  decrease  attending  an  in- 
finitesimal change  in  volume  or  pressure  at  a  constant  temperature  is 
therefore  given  by  the  expression: 

-dF=-vdp. 
For  a  finite  change  in  volume  or  pressure  the  corresponding  expression 

is: 

/*£i 

-AF=  I     vdp. 
Jpt 

This  equation  is  a  general  one,  applicable  to  gaseous,  liquid,  and 
solid  systems.  In  order  to  carry  out  the  indicated  integration,  the 
functional  relation  between  the  pressure  and  volume  of  the  system 
at  constant  temperature  must  be  known. 

Prob.  7.  —  Free-Energy  Decrease  Attending  the  Expansion  of  Gases.  — 
a.  Derive  an  expression  for  the  free-energy  decrease  attending  the 
change  in  state  of  N  mols  of  a  perfect  gas  when  at  the  temperature  T 
its  volume  and  pressure  change  from  v\  and  pi  to  v2  and  pz.  b.  Derive  by 
integration  an  expression  for  the  decrease  of  the  free  energy  of  N  mols 
of  a  gas  when  at  T  its  pressure  changes  from  pi  to  p2,  provided  the 
pressures  are  moderate  so  that  its  pressure-volume  relations  are  ex- 
pressed with  sufficient  accuracy  by  the  equation  pv=NRT(i+ap) 
given  in  Art.  15.  c.  Calculate  by  this  expression  the  value  in  joules  of 
the  free-energy  decrease  attending  the  expansion  of  i  mol  of  CO2  from 
4  atm.  to  i  atm.  at  o°-,  for  which  substance  at  o°  the  deviation-co- 
efficient a  has  the  value  —0.0068  per  atmosphere.  Ans.  c,  3100. 


FREE  ENERGY  OF  PHYSICAL  CHANGES  231 

*Prob.  8.  —  Free-Energy  Decrease  Attending  the  Compression  of 
Liquids.  —  The  change  in  volume  v  of  liquid  or  solid  substances  with 
the  pressure  p  can  commonly  be  approximately  expressed  by  the  equa- 
tion V=VQ(I  —  kp),  where  v0  is  the  volume  at  zero  pressure  and  k  is  a 
quantity,  called  the  compression-coefficient,  which  varies  with  the  sub- 
stance and  with  the  temperature,  but  may  ordinarily  be  regarded  as 
constant  for  variations  of  pressure  up  to  30  to  50  atm.  a.  Derive  an 
expression  for  the  decrease  in  free  energy  of  such  a  substance  when  the 
pressure  on  it  changes  from  pi  to  pz.  b.  Find  the  free-energy  decrease 
in  joules  attending  the  compression  from  i  atm.  to  50  atm.  at  20°  of 
10  g.  of  water,  whose  compression-coefficient  at  20°  is  0.000048  when 
the  pressure  is  in  atmospheres,  and  whose  density  at  20°  and  i  atm. 
is  0.998.  Ans.  b,  49.7. 

138.  Free-Energy  Changes  Attending  the  Transfer  of  Substances 
between  Solutions  of  Different  Concentrations.  —  As  shown  in  the 
problems  below,  there  may  readily  be  derived  the  following  expressions 
for  the  free-energy  decrease  which  attends  the  transfer  at  the  tem- 
perature T  of  that  quantity  of  a  substance  which  is  N  mols  in  the 
state  of  a  perfect  gas  from  an  infinite  quantity  of  a  solution  in  which 
its  vapor-pressure  is  pi,  its  mol-fraction  xi,  and  its  molality  or  molal 
concentration  ci,  into  an  infinite  quantity  of  another  solution  in  which 
its  vapor-pressure  is  p*,  its  mol-fraction  x*,  and  its  molality  or  molal 
concentration^: 

(i) 


(3) 

The  conditions  under  which  each  of  these  equations  is  exact  will  be 
evident  from  its  derivation.  Equation  (i)  holds  true  whatever  be 
the  mol-fractions  or  concentrations;  but  it  involves  the  assumption 
that  the  vapor  conforms  to  the  perfect-gas  law.  Equation  (2)  holds 
true  when,  in  conformity  either  with  Raoult's  law  or  with  Henry's 
law,  the  vapor-pressures  are  proportional  to  the  mol-fractions.  Equa- 
tion (3)  holds  true  when,  in  conformity  with  Henry's  law,  the  vapor- 
pressures  are  proportional  to  the  mol-fractions  and  when  the  latter 
are  small  enough  to  be  substantially  proportional  to  the  concentra- 
tions; it  is  the  expression  commonly  employed  when  a  solute  is  trans- 
ferred  from  one  dilute  solution  to  another. 


232      PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

Derivation  of  the  Free-Energy  Equations.  — 

Prob.  g.  —  a.  Formulate  an  exact  expression  (not  involving  the 
perfect-gas  law)  for  the  free-energy  decrease  attending  the  introduction 
at  the  temperature  T  of  m  grams  of  a  pure  substance  at  a  pressure  equal 
to  its  vapor-pressure  pQ  into  an  infinite  quantity  of  a  solution  in  which 
the  substance  has  a  vapor-pressure  p.  Note  that  this  change  in  state 
can  be  brought  about  by  the  following  reversible  process:  vaporize  the 
m  grams  of  the  substance  at  the  temperature  T  under  a  pressure  pQ\ 
change  the  pressure  of  this  vapor  to  p;  and  at  this  pressure  condense  the 
vapor  into  the  solution,  b.  Derive  from  this  expression  an  equation 
which  holds  true  when  the  vapor  conforms  to  the  perfect-gas  law. 
c.  Formulate  an  expression  for  the  free-energy  decrease  attending  the 
transfer  of  N  mols  of  a  substance  from  an  infinite  quantity  of  a  solution 
in  which  its  vapor-pressure  is  pi  into  an  infinite  quantity  of  a  solution 
in  which  its  vapor-pressure  is  pz. 

Prob.  10.  —  Show  how  the  equation  formulated  in  Prob.  gc  may  be 
modified  so  as  to  contain  the  mol-fractions  x\  and  xz  of  the  substance 
in  the  two  solutions,  a,  when  the  vapor-pressures  p\  and  pz  conform  to 
Raoult's  law;  6,  when  these  vapor-pressures  conform  to  Henry's  law. 
c.  State  in  the  case  of  a  solution  consisting  of  two  substances  A  and  B 
the  conditions  of  composition  under  which  Raoult's  law  and  under  which 
Henry's  law  (and  therefore  under  which  the  corresponding  free-energy 
expressions)  hold  true  approximately,  d.  Show  that  when  the  mol- 
fractions  xi  and  x2  are  small,  their  ratio  may  be  replaced  by  the  ratio 
of  the  molalities  or  molal  concentrations,  which  signify  (Art.  35)  the 
number  of  mols  of  the  solute  per  1000  grams  or  1000  ccm.,  respectively, 
of  the  solvent. 

Since  Henry's  law  applies  to  the  concentrations  of  a  definite  chemical 
substance,  it  is  evident  that  equation  (3)  must  be  applied  separately 
to  the  transfer  of  each  chemical  substance,  not  to  the  transfer  of  the 
substance  as  a  whole,  if  it  exists  in  the  solution  as  two  or  more  different 
chemical  substances  as  is  the  case  with  H+C1~  or  any  other  partially 
ionized  substance  in  aqueous  solution,  or  with  acetic  acid  in  benzene 
solution  in  which  it  exists  as  C^H-A  and  (CzH-tOz)*. 

Although  equation  (3)  was  derived  by  the  consideration  of  a  process 
involving  the  vaporization  of  the  solute  and  with  the  aid  of  the  assump- 
tion that  the  pressure  of  the  vapor  was  small  enough  to  conform  to 
the  perfect-gas  law,  yet  that  equation  relates  only  to  the  transfer 
of  the  substance  from  one  solution  to  another;  and  it  would  be  re- 
markable if  its  validity  depended  on  whether  the  substance  were 
volatile  or  on  how  large  its  vapor-pressure  might  be.  In  fact,  it  can 
be  shown,  by  deriving  equation  (3)  through  a  consideration  of  an 
osmotic  process  of  transferring  the  substance  from  one  solution  to  the 


FREE  ENERGY  OF  PHYSICAL  CHANGES  233 

other,  that  the  equation  is  exact,  provided  only  that  the  substance 
behaves  as  a  perfect  solute,  as  shown  by  its  conformity  to  the  osmotic- 
pressure  equation  PvQ=  NRT  considered  in  Art.  52. 

Applications  of  the  Free  Energy  Equations.  — 

Prob.  ii.  —  a.  Calculate  the  free-energy  decrease  attending  the  trans- 
fer at  20°  of  iNH3  from  an  infinite  quantity  of  a  solution  of  the  composi- 
tion i  NH3.8|H2O  in  which  its  vapor-pressure  is  80  mm.  into  an  infinite 
quantity  of  a  solution  of  the  composition  iNH3.2iH2O  in  which  its 
vapor-pressure  is  27  mm.  b.  Calculate  the  free-energy  decrease 
attending  the  dissolving  at  20°  of  iNH3  at  i  atm.  in  an  infinite  quantity 
of  a  solution  of  the  composition  iNH3.2iH2O. 

Note.  —  Throughout  this  chapter  free-energy  values  are  to  be  ex- 
pressed in  calories. 

Prob.  12.  —  Calculate  the  free-energy  decrease  attending  the  transfer 
at  20°  of  iNH3  from  an  infinite  quantity  of  o.i  formal  NH4OH  solution 
into  an  infinite  quantity  of  o.ooi  formal  NH4OH  solution.  The  am- 
monium hydroxide  is  1.3%  ionized  in  the  o.i  formal  and  12.5%  ionized 
in  the  o.ooi  formal  solution,  and  the  remaining  ammonium  hydroxide 
is  y  per  cent  dissociated  into  NH3  and  H2O  in  each  solution. 

Prob.  13.  —  The  transfer  at  the  temperature  T  of  one  formula-weight 
of  HCO2H  from  an  infinite  quantity  of  a  dilute  solution  in  which  its 
formality  is  c\  and  its  ionization  is  71  into  an  infinite  quantity  of  another 
dilute  solution  in  which  its  formality  is  c2  and  its  ionization  is  yt  can 
be  brought  about  by  so  transferring  either  i  mol  of  HCO2H  or  one  mol 
of  H+  and  one  mol  of  HCO2~.  a.  Formulate  an  expression  for  the  free- 
energy  decrease  attending  each  of  these  processes,  b.  Show  that 
one  of  these  expressions  can  be  derived  from  the  other  with  the  aid  of 
the  mass-action  equation  for  the  ionization  of  the  salt.  c.  Calculate  the 
free-energy  decrease  by  both  expressions  for  the  case  that  the  tempera- 
ture is  25°,  the  concentrations  are  o.i  and  o.ooi  formal,  and  the  ioni- 
zations  are  0.045  and  0.360. 

Prob.  14.  —  Formulate  an  expression  for  the  free-energy  decrease 
attending  the  transfer  of  iK2SO4  at  25°  from  a  o.i  formal  solution  to  a 
o.ooi  formal  solution,  assuming  that  the  salt  is  completely  ionized  and 
that  the  ions  act  as  perfect  solutes. 

Note.  —  The  assumptions  that  largely  ionized  substances  are  com- 
pletely ionized  and  that  their  ions  are  perfect  solutes  are  to  be  made, 
unless  otherwise  stated,  also  in  all  later  problems. 

*Owing  to  the  fact  that  ions  at  even  small  concentrations  are  im- 
perfect solutes,  the  free-energy  decrease  attending  the  transfer  of 
largely  ionized  substances  is  expressed  only  as  a  rough  approximation 
by  equation  (3).  An  expression  exact  at  any  concentration  may, 
however,  be  formulated  with  the  aid  of  the  concept  of  activity  pre- 
sented in  Art.  113.  Namely,  it  follows  from  the  proportionality 


234   PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

between  the  activity  of  any  definite  substance  in  a  solution  and  its 
vapor-pressure  (assuming  the  vapor  to  be  a  perfect-gas)  that  the 
ratio  of  the  activities  of  a  substance  in  two  solutions,  or  in  general  in 
two  phases,  may  be  substituted  for  the  ratio  of  the  pressures  in  equa- 
tion (i).  Hence,  for  the  transfer  of  N  mols  of  a  chemical  substance 
at  T  from  a  solution  or  phase  in  which  its  activity  is  a\  to  a  solution  or 
phase  in  which  its  activity  is  02  the  free- energy  decrease  is: 

-AF  =  NRT  log  —  -  (4) 

#2 

*This  equation  has  practical  significance,  not  so  much  because  it 
furnishes  a  means  of  evaluating  free  energies,  as  because  it  supple- 
ments the  primary  definition  of  activity  (given  by  the  equation  a\ja^  = 
pi/pz)  by  a  secondary  definition  which  affords  another  method  of 
determining  the  relative  activities  of  a  substance  in  different  solutions. 
Namely,  it  enables  activities  and  activity-coefficients  to  be  derived 
from  the  free-energy  decrease  attending  the  transfer  of  the  substance 
in  cases  where  this  decrease  can  be  obtained  from  other  measure- 
ments than  those  of  vapor-pressure.  It  will  be  shown  in  Art.  146 
that  this  decrease  can  be  computed  from  the  electromotive  force  of 
certain  types  of  voltaic  cells;  and  applications  of  equation  (4)  will 
there  be  considered. 

139.  Free-Energy  Changes  Attending  the  Transfer  of  Substances 
between  Different  Phases.  —  When  the  change  in  state  consists  in 
the  transfer  of  a  substance  from  one  phase  to  another  phase  at  the 
temperature  and  pressure  at  which  the  two  phases  are  in  equilibrium, 
the  free-energy  decrease  is  zero.  Changes  in  state  of  this  type  are  the 
transition,  fusion,  or  vaporization  of  a  solid  substance,  and  the  freezing 
or  vaporization  of  a  liquid  substance,  provided  the  two  phases  are  at 
the  temperature  and  pressure  at  which  they  are  in  equilibrium.  This 
principle  follows  from  the  general  expression  for  free-energy  decrease, 
— AF  =  W*  —  A(/rc>),  and  from  the  fact  that  this  type  of  change  can  be 
made  to  take  place  reversibly  at  the  given  pressure,  so  that  the  maxi- 
mum work  WR  is  simply  equal  to  that  pressure  times  the  volume- 
increase,  that  is,  to  p(v2—vi),  which  is  identical  with  A(pv). 

Prob.  15.  —  Vaporization,  Fusion,  and  Transition  in  Relation  to  Free 
Energy.  —  Name  a  condition  of  temperature  and  pressure  at  which  the 
free  energies  are  equal  and  a  condition  under  which  they  are  not  equal, 
a,  for  water  as  a  liquid  and  as  a  gaseous  phase;  b,  for  water  as  a  solid 
and  as  a  liquid  phase;  c,  for  sulfur  in  the  rhombic  and  the  monoclinic 
forms  (see  Art.  115). 


FREE  ENERGY  OF  PHYSICAL  CHANGES  235 

Another  case  of  this  type  is  the  transfer  of  a  substance  from  a  solid 
phase  to  a  solution  saturated  with  respect  to  that  substance  at  a 
definite  temperature  and  pressure.  In  order  that  this  change  in  state 
may  take  place  reversibly,  the  solid  substance  is  made  to  dissolve  in  an 
infinite  volume  of  its  saturated  solution  by  increasing  or  decreasing 
the  pressure  on  the  two  phases  by  an  infinitesimal  amount.  The  work 
produced  in  this  process  is  again  simply  equal  to  the  pressure  times 
the  change  in  volume  (p  A»),  and  the  free-energy  decrease  is  therefore 
zero. 

Prob.  16.  —  Solution  in  Relation  to  Free  Energy.  —  Find  the  maximum 
work  and  the  free-energy  decrease  attending  the  melting  of  iH2O  as  ice 
at  —1.69°  into  a  0.5  formal  NaCl  solution  at  its  freezing-point  —1.69° 
at  i  atm.  The  molal  volume  of  ice  is  19.6  can.,  and  the  increase  in 
volume  of  an  infinite  quantity  of  the  solution  when  iH2O  is  added  to 
it  is  1 8.0  ccm. 

Still  another  case  of  this  type  is  the  transfer  at  a  definite  temperature 
of  a  substance  from  one  liquid  phase  to  another  liquid  phase,  thus  from 
a  solution  of  it  in  one  solvent  to  a  solution  of  it  in  another  solvent  not 
miscible  with  the  first  solvent.  Provided  the  substance  is  present  in 
the  two  phases  at  the  equilibrium  concentrations,  the  free-energy 
decrease  attending  such  a  transfer  is  zero,  as  shown  in  Prob.  17. 

Prob.  17.  —  Distribution  in  Relation  to  Free  Energy.  —  A  c\  formal 
solution  of  NH3  in  chloroform  is  in  equilibrium  with  a  c2  formal  solution 
of  NH3  in  water  at  T.  Show  that  the  free-energy  decrease  is  zero  for  the 
transfer  of  iNH3  from  an  infinite  quantity  of  the  chloroform  solution 
into  an  infinite  quantity  of  the  water  solution,  by  considering  a  reversible 
process  involving  the  vapor  phase. 

When  the  substance  in  one  phase  at  any  definite  temperature  is  not 
in  equilibrium  with  the  same  substance  in  the  other  phase,  the  transfer 
of  it  from  one  phase  to  the  other  is  attended  by  an  increase  or  decrease 
in  its  free-energy.  In  determining  this  free-energy  change,  a  process 
must  as  usual  be  conceived  by  which  the  transfer  takes  place  reversibly, 
and  the  work  attending  this  reversible  process  must  then  be  evaluated. 
This  is  illustrated  by  the  following  problems. 

Free-Energy  Decrease  Attending  the  Conversion  of  Unstable  into  Stable 
Phases.  — 

Prob.  18.  —  Find  the  free-energy  decrease  that  attends  the  transition 
at  105°  of  iS  from  the  rhombic  into  the  monodinic  form,  each  being  at  a 
pressure  equal  to  its  vapor-pressure,  referring  to  the  figure  of  Art.  115 
for  the  data  needed  and  assuming  that  the  molecular  formula  of  the 
sulfur  vapor  is  SB. 


236   PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

Prob.  19.  —  Find  the  free-energy  decrease  attending  the  transition  of 
32.06  g.  of  monoclinic  to  rhombic  sulfur  at  25°  from  the  facts  that  at 
25°  the  solubility  in  benzene  of  the  monoclinic  form  is  23.2  g.  and  that 
of  the  rhombic  form  is  18.2  g.  per  liter  of  benzene,  and  that  the  molecular 
formula  of  sulfur  in  benzene  has  been  shown  by  molecular-weight 
determinations  to  be  S8.  Consider  a  reversible  process  that  involves 
dissolving  the  monoclinic  form  in  its  saturated  solution,  transferring 
the  sulfur  from  this  solution  to  one  saturated  with  respect  to  the  rhom- 
bic form,  and  causing  the  rhombic  form  to  precipitate.  Ans.  18  cal. 

*Prob.  20.  —  a.  Find  the  free-energy  decrease  that  attends  the  change 
of  iH2O  from  the  state  of  liquid  at  — 1°  and  4.255  mm.  (its  vapor- 
pressure  at  —i°)  to  the  state  of  ice  at  — 1°  and  4.215  mm.  (its  vapor- 
pressure  at  —i°)  by  considering  a  reversible  process  similar  to  that  of 
Prob.  9.  b.  Find  the  free-energy  decrease  attending  the  same  change 
in  state,  by  considering  a  reversible  process  in  which  the  water  and  ice 
are  brought  into  equilibrium  at  — 1°  by  compression.  Ice  melts  at 
—  i°  when  the  pressure  is  130  atm.  The  mean  compression-coefficient 
(see  Prob.  8)  at  o°  between  zero  pressure  and  this  pressure  is  0.000051 
for  water  and  0.000034  for  ice. 

The  foregoing  considerations  may  be  summed  up  in  the  following 
statements: 

(1)  The  free  energies  of  a  substance  in  two  different  phases  at  any 
definite  temperature  are  equal  when  the  pressure  or  concentration 
is  that  at  which  there  is  equilibrium. 

(2)  When  the  pressure  or  concentration  is  not  that  corresponding  to 
equilibrium,  the  substance  in  the  unstable  state  has  the  greater  free 
energy;  and  namely,  a  free  energy  greater  by  the  amount  given  by  the 
expressions  NRT  log(/>i/^2)  and  NRT  log  (si/sz),  in  which  pi  and  pz 
represent  the  vapor-pressures  of  the  substance  in  the  unstable  and 
stable  states,  respectively,  and  Si  and  sz  represent  its  solubilities  in  any 
solvent  in  the  unstable  and  stable  states,  respectively.    This  second 
principle  evidently  corresponds  to  the  principles  stated  in  Art.  115 
that  the  unstable  form  of  a  substance  has  the  greater  vapor-pressure 
and  the  greater  solubility  in  any  solvent. 

These  principles  show  that  equality  of  the  free  energies  of  a  sub- 
stance in  two  phases  is  a  criterion  of  equilibrium;  and  that,  when  the 
free  energies  are  not  equal,  a  change  tends  to  take  place  in  that  direc- 
tion in  which  there  is  an  actual  decrease  in  the  free-energy. 

*Since  the  activity  of  the  substance  in  any  two  phases  is  by  defini- 
tion proportional  to  its  vapor-pressure  (Art.  113),  it  follows  that  the 
activity  of  a  substance  is  the  same  in  different  phases  which  are  in 


FREE   ENERGY  OF  PHYSICAL  CHANGES  237 

equilibrium  with  one  another.  For  example,  when  solid  iodine  is 
shaken  with  water  and  chloroform  at  25°  in  a  closed  vessel  till  equi- 
librium is  reached,  the  activity  of  the  iodine  is  the  same  in.  the  solid, 
the  gaseous,  and  the  two  liquid  phases.  This  is  a  fundamental 
principle  which  is  often  made  use  of  in  applications  of  the  mass-action 
law.  Thus  in  the  mass-action  expression  (HI)2/(H2S)(I2)  =  K,  for 
the  equilibrium  of  the  reaction  H2S  +  I2  =  2HI+S(s)  in  aqueous 
solution,  the  concentrations  of  the  H2S  and  HI  can  be  replaced  by 
their  vapor-pressures,  and  that  of  the  I2  can  be  replaced  by  its  con- 
centration in  a  chloroform  phase  in  equilibrium  with  the  aqueous 
solution.  Such  substitutions  are  useful,  either  when  the  substance 
is  an  imperfect  solute  in  the  aqueous  solution,  or  when  its  pressure 
in  the  gas  phase  or  its  concentration  in  another  solvent  can  be  more 
easily  or  more  accurately  determined  than  its  concentration  in  the 
aqueous  solution. 

*In  case  there  is  not  equilibrium  it  follows  from  the  relation  between 
free-energy  decrease  and  activities  derived  in  Art.  138  that  the  ratio 
of  the  activities  of  the  substance  in  the  two  phases  is  given  by  the 

equation: 

fli       -AF 

10g  02  ~  NRT 

Qualitatively  this  signifies  that  the  activity  of  a  substance  is  greater 
in  the  unstable  state;  for  this  state  always  passes  over  into  the  stable 
state  with  a  positive  free-energy  decrease. 


238      PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 


m.    FREE-ENERGY  CHANGES  ATTENDING  CHEMICAL  CHANGES 

140.  The  Free-Energy  Equation  for  Chemical  Changes  between 
Perfect  Gases.  —  An  expression  for  the  free-energy  decrease  attend- 
ing a  chemical  change  between  perfect  gaseous  substances  in  terms  of 
the  equilibrium-constant  for  that  change  can  be  derived  by  the  follow- 
ing considerations. 

Consider  any  chemical  reaction  0A  +  6B.  .  =eE+fF.  .between 
the  gaseous  substances  A,  B,.  .  E,  F,.  . ;  and  consider  a  change  in  state 
which  consists  in  the  conversion  at  the  temperature  T  of  a  mols  of  A, 
b  mols  of  B,.  .at  pressures  pA',  p*r  .  into  e  mols  of  E  and /mols  of  F,.  . 
at  pressures  £E',  py.  .  This  change  in  state  is  briefly  expressed  by 
the  equation: 

a  A  (at  p/)  +  6B  (at  £/)•  . .  =  «E  (at  p*9)  +/F  (at  £/). . . 

This  reaction  would  take  place  continuously  in  one  direction  or  the 
other  (unless  the  pressures  happened  to  be  those  corresponding  to 
equilibrium)  if  infinite  quantities  of  the  gases,  each  at  its  specified 
pressure,  were  present  together  in  a  mixture;  and  the  specified  change 
in  state  would  be  realized  if  the  period  during  which  the  specified 
number  of  mols  reacted  were  considered.  However,  the  change  in 
state  would  not  be  taking  place  reversibly,  since  no  mechanism  is 
provided  by  which  the  work  which  the  change  is  capable  of  producing 
would  be  actually  produced.  It  is  therefore  necessary  to  conceive 
of  some  process  by  which  the  change  in  state  can  be  made  to  take 
place  reversibly.  Such  a  process  will  now  be  described. 

In  the  process  an  apparatus  like  that  shown  in  the  figure  will  be 
employed.  This  apparatus  consists  of  a  reservoir  and  of  a  number  of 
cylinders  which  communicate  with  it  through  walls,  each  of  which  is 
permeable  for  only  one  of  the  gases.  These  walls  can  be  replaced  at 
will  by  impermeable  ones.  The  cylinders,  which  serve  to  hold  the 


Gas 
A 


Gas 
B 


Gas 
E 


Gas 
F 


Equilibrium  Mixture  of  A ,  B,  E,  and  F 


FREE  ENERGY  OF  CHEMICAL  CHANGES      239 

separate  gases,  are  provided  with  weighted  frictionless  pistons.  The 
reservoir  contains  a  mixture  of  the  gases  at  any  partial  pressures 
PA,  ps,-  -pE,  PY,-  -at  which  the  chemical  change  is  in  equilibrium. 

Starting  now  with  a  mols  of  A,  b  mols  of  B,.  .  at  pressures  pA',  p*  .  . 
in  their  respective  cylinders,  consider  the  following  process  to  be 
carried  out  at  the  temperature  T  under  reversible  conditions,  always 
keeping  the  external  pressure  on  any  piston  equal  within  an  infinitesi- 
mal amount  to  that  of  the  gas  beneath. 

1.  The  cylinders  being  closed  temporarily  by  impermeable  walls, 
raise  or  lower  the  pistons  above  the  gases  A,  B,.  .till  the  pressures 
change  from  PA,  p*r  -to  those,  pA,  ps.  .,  prevailing  in  the  equilib- 
rium-mixture. 

2.  Replace  the  impermeable  walls  by  semipermeable  ones,  and  force 
the  a  mols  of  A,  b  mols  of  B,.  .into  the  reservoir,  at  the  same  time 
drawing  out  into  the  other  cylinders  at  the  pressures  pE,  pf,.  .pre- 
vailing in  the  equilibrium-mixture,  the  e  mols  of  E,  /  mols  of  F,.  ; 
which  must  form  spontaneously  out  of  A,  B,.  .in  order  that  the  equi- 
librium in  the  mixture  may  be  maintained. 

3.  Replace  the  semipermeable  walls  under  the  gases  E,  F,.  .by 
impermeable  ones,  and  raise  or  lower  the  pistons  above  the  gases 
E,   F,.  .till   the   pressures   change  from   the   equilibrium-pressures 
pz,  pt,-  -to  the  final  pressures  p*',  />/.  . 

By  formulating,  as  is  done  in  Prob.  21  expressions  for  the  free- 
energy  decrease  attending  each  of  the  steps  in  this  process  and  sum- 
ming up  the  three  values,  there  is  obtained  in  the  case  that  all  the 
pressures  involved  are  so  small  that  they  conform  substantially  to 
the  perfect-gas  law  the  following  expression  for  the  free-energy  decrease 
attending  an  isothermal  chemical  change  between  gaseous  substances: 


Prob.  21.  —  Derivation  and  Significance  of  the  Free-Energy  Equation. 
a.  Formulate  expressions  for  the  free-energy  decrease  attending  each 
of  the  three  steps  in  the  process  described  in  the  preceding  text.  b.  By 
combining  these  expressions  derive  the  free-energy  equation  there 
given,  c.  State  explicitly  the  change  in  state  to  which  this  equation 
applies  and  the  difference  in  the  significance  of  the  pressures  in  the 
two  logarithmic  terms;  and  name  a  familiar  quantity  that  may  be 
substituted  in  one  of  these  terms,  d.  Give  the  form  which  the  equation 
assumes  when  the  initial  and  final  partial  pressures  are  all  unity. 


240      PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

Prob.  22.  —  Evaluation  of  the  Free-Energy  Decrease.  —  a.  Calculate 
the  free-energy  decrease  in  calories  attending  at  2000°  the  change 
2H2(o.i  atm.)+O2(o.5  atm.)  =  2H20(i  atm.)  from  the  fact  that  water- 
vapor  at  2000°  and  i  atm.  is  1.85%  dissociated  into  hydrogen  and 
oxygen,  b.  Calculate  also  the  free-energy  decrease  at  2000°  when  the 
initial  pressures  of  the  hydrogen  and  the  oxygen  (as  well  as  the  final 
pressure  of  the  water- vapor)  are  each  i  atm.  Ans.  b,  57,120. 

Prob.  23.  —  Derivation  of  the  Mass- Action  Law.  —  Derive  the  mass" 
action  law  expressed  in  terms  of  pressures  by  considering  in  the  deriva- 
tion of  the  free-energy  equation  that  the  same  change  in  state  is  brought 
about  by  two  reversible  processes  of  the  type  described  in  the  text 
differing  in  the  respect  that  different  equilibrium  mixtures  are  contained 
in  the  reservoir. 

Prob.  24.  —  Relation  of  the  Free-Energy  Change  to  the  Tendency  of 
the  Chemical  Change  to  Take  Place.  —  a.  Show  by  the  mass-action  law 
that;  when  the  pressures  of  the  substances  in  a  gaseous  mixture  are 
such  that  the  free-energy  decrease  attending  the  reaction  as  computed 
by  the  free-energy  equation  has  a  positive  value,  the  reaction  as  written 
takes  place  spontaneously  from  left  to  right;  and  that,  when  the  pres- 
sures are  such  that  the  computed  free-energy  decrease  has  a  negative 
value,  the  reaction  as  written  takes  place  spontaneously  from  right  to 
left;  and  therefore  that  the  reaction  always  takes  place  in  that  direction 
in  which  there  is  a  positive  free-energy  decrease,  b.  Show  that  equi- 
librium prevails  when  the  pressures  of  the  substances  in  a  gaseous 
mixture  are  such  that  the  computed  free-energy  decrease  is  zero. 

The  conclusions  reached  in  the  preceding  problem  afford  another 
example  of  the  following  general  principles,  already  shown  in  Art.  139 
to  apply  to  the  transfer  of  substances  between  phases: 

(1)  When  at  any  definite  temperature  the  pressures  or  concentra- 
tions of  the  substances  present  in  a  system  are  such  that  a  change  in  the 
state  would  be  attended  by  zero  change  in  free-energy,  the  substances 
involved  in  that  change  are  in  equilibrium  with  each  other.    That  is, 
the  condition  —  AF  =  o  is  a  criterion  of  the  equilibrium  of  the  sub- 
stances involved  in  any  change  in  state. 

(2)  When  at  any  definite  temperature  the  pressures  or  concentra- 
tions are  such  that  a  change  in  the  state  of  the  system  would  be 
attended  by  a  change  in  its  free-energy,  the  change  tends  to  take  place 
in  that  direction  in  which  the  free-energy  actually  decreases.     That 
is,  a  system  can  spontaneously  change  its  state  only  in  that  direction 
for  which  —  AF  is  positive. 

141.  The  Free-Energy  Equation  for  Chemical  Changes  between 
Perfect  Solutes.  —  It  can  be  shown,  as  is  done  in  Prob.  25,  that  the 


FREE  ENERGY  OF  CHEMICAL  CHANGES      241 

free-energy  decrease  attending  at  the  temperature  T  the  chemical 
change: 

a  A(at  c>!)  +  &B(at  c»').  -  =  *E(at  CE')  +/F(at  cv').  .  , 
where  A,  B.  .E,  F,.  .are  dissolved  substances  at  the  concentrations 
CA',  c*,.  .cE',  cv'..,  which  are  so  small  that  the  substances  behave  as 
perfect  solutes,  is  given  by  the  equation: 


In  this  equation,  which  is  obviously  closely  analogous  to  that  for 
gaseous  substances,  the  quantities  CA,  c*,.  .CE,  cr,.  .are  any  small 
concentrations  at  which  the  substances  are  in  equilibrium. 

Prob.  25.  —  Derivation  of  the  Free-Energy  Equation.  —  Derive  the 
free-energy  equation  given  in  the  preceding  text  by  considering  that 
all  the  solutes  are  volatile  and  that  the  change  in  state  is  brought  about 
by  vaporizing  A,  B,.  .  out  of  their  solutions,  converting  them  into  E, 
F,  .  .  in  the  gaseous  state  by  the  process  described  in  the  text  of  Art.  140, 
and  condensing  E,  F,  .  .into  their  solutions;  each  of  these  changes  being 
carried  out  under  equilibrium-conditions  so  that  it  may  be  reversible. 
Represent  by  pA',  PA,.  .pE,pE.  .the  partial  vapor-pressures  correspond- 
ing to  the  concentrations  CA  ',  CA,  .  .  CK,  CE,  .  .  ,  and  assume  that  in  ac- 
cordance with  Henry's  law  the  pressures  and  concentrations  are  pro- 
portional. 

Prob.  26.  —  Evaluation  of  the  Free-Energy  Decrease.  —  Calculate 
the  free-energy  decrease  that  attends  at  25°  the  chemical  change 
NH4+(at  o.i  f.)  +  CN-(at  o.i  f.)  =  NH3(at  o.i  f.)  +  HCN(at  o.i  f.), 
assuming  that  NH4CN  in  o.i  formal  solution  at  25°  is  47%  dissociated 
into  NH3  and  HCN,  and  that  the  remainder  of  the  salt  is  completely 
ionized. 

Although  this  free-energy  equation  was  derived  above  under  the 
assumptions  that  the  solutes  are  volatile  and  that  their  concentrations 
and  vapor-pressures  are  so  small  that  Henry's  law  is  applicable,  it 
can  be  shown  that  this  equation  (like  equation  (3)  of  Art.  138)  is 
valid  provided  only  that  the  concentrations  are  so  small  that  the 
solutes  conform  to  the  laws  of  perfect  solutions. 

The  mass-action  law  for  perfect  solutes  can  be  derived  from  this 
free-energy  equation  just  as  the  mass-action  law  for  perfect  gases 
was  derived  in  Prob.  23  from  the  free-energy  equation  of  the  preceding 
article.  The  validity  of  the  mass-action  law  therefore  involves  only 
that  of  the  laws  of  perfect  solutions  or  of  perfect  gases;  and  it  will  hold 
true  when  applied  to  any  definite  chemical  change  with  a  degree  of 


242      PRODUCTION  OF  WORK  BY  CHEMICAL  CHANGES 

exactness  corresponding  to  that  with  which  the  substances  involved 
conform  under  the  given  conditions  of  concentration  or  pressure  to 
the  laws  of  perfect  solutions  or  of  perfect  gases. 

142.  The  Free-Energy  Equations  for  Chemical  Changes  between 
Solid  Substances  and  Perfect  Gases  or  Perfect  Solutes.  —  When 
pure  solid  substances  are  involved  in  a  chemical  change  with  gaseous 
or  dissolved  substances  it  is  not  necessary  to  include  in  the  free-energy 
equation  their  pressures  or  concentrations;  for  these  would  obviously 
have  the  same  value  (that  of  the  vapor-pressure  or  solubility  of  the 
solid)  in  the  two  logarithmic  terms  corresponding  to  the  equilibrium 
conditions  and  to  the  initial  and  final  conditions.  The  same  is  true 
of  the  pressure  of  a  pure  liquid  when  it  is  involved  in  a  chemical  change 
with  gaseous  substances  that  are  not  much  soluble  in  it,  as  in  the 
formation  of  liquid  water  from  hydrogen  and  oxygen. 

Evaluation  of  the  Free-Energy  Decrease.  — 

Prob.  27.  —  Calculate  from  the  data  of  Prob.  39,  Art.  no,  the  free- 
energy  decrease  attending  at  25°  the  chemical  change: 

NH4SH(s)  =NH3(i  atm.)+H2S(i  atm.). 

Prob.  28.  —  Calculate  from  the  data  of  Prob.  41,  Art.  no,  the  free- 
energy  decrease,  attending  at  357°  the  change  in  state: 
Hg(l)  +!0,(i  atm.)  =HgO(s).  , 

Prob.  29.  —  Calculate  from  the  data  of  Prob.  54,  Art.  112,  the  free- 
energy  decrease  attending  at  25°  the  chemical  change: 

AgSCN(s)+Br~  (o.i  f.)=AgBr(s)+SCN-  (o.i  f.). 


CHAPTER  XI 

THE  PRODUCTION  OF  WORK  FROM  ISOTHERMAL 
CHANGES  BY  ELECTROCHEMICAL  PROCESSES 


I.    CHANGES   IN   STATE  AND  IN  FREE  ENERGY  IN  VOLTAIC  CELLS 

143.  Introduction.  —  It  has  been  shown  in  the  preceding  chapter 
that  the  free-energy  decrease  attending  changes  in  state  can  be  derived 
from  the  consideration  of  certain  mechanical  processes  by  which  the 
changes  in  state  take  place  reversibly;  and  thereby  important  rela- 
tions between  free  energy  and  the  equilibrium  conditions  of  chemical 
reactions  have  been  deduced.     In  this  chapter  it  is  shown  that  electro- 
chemical processes  afford  another  means  of  carrying  out  changes  in 
state  reversibly  and  hence  of  evaluating  the  free-energy  decrease 
attending  them.  The  changes  in  state  occurring  in  voltaic  cells  are 
first  considered  (in  Art.  144).    The  electromotive  force  of  voltaic 
cells,  the  production  of  work  by  them,  and  the  corresponding  free- 
energy  decrease  are  then  treated  (hi  Art.  145)  in  their  general  aspects. 
In  the  following  articles  these  general  principles  are  applied  to  the 
determination  of  the  electromotive  force  and  of  the  separate  electrode- 
potentials  of  special  types  of  cells. 

144.  Changes  in  State  in  Voltaic  Cells.  —  A  change  in  state  can  be 
made  to  yield  electrical  energy  when  it  can  be  brought  about  by  a 
process  of  reduction  occurring  in  one  place  and  a  process  of  oxidation 
occurring  in  another  place;  for  it  is  an  inherent  characteristic  of  reduc- 
tion processes  that  they  are  attended  by  a  liberation  of  positive 
electricity  or  by  an  absorption  of  negative  electricity,  and  of  oxidation 
processes  that  they  are  attended  by  the  opposite  electrical  effects. 
Thus  the  reduction  of  copper-ion  to  metallic  copper  may  be  repre- 
sented by  the  equation  Cu++  =  Cu  +  20;  and  the  oxidation  of  metallic 
zinc  to  zinc-ion,  by  the  equation  Zn  +  2©  =Zn++,  where  the  symbol  © 
denotes,  as  in  Art.  63,  one  positive  electron  or  one  faraday  of  positive 
electricity.     If  these  two  processes  occur  at  the  same  place,  as  is  the 
case  when  metallic  zinc  is  placed  in  a  copper-ion  solution,  no  electrical 
effect  is  observed.     If,  however,  the  zinc  is  placed  in  a  zinc-ion 
solution  and  the  copper  in  a  copper-ion  solution,  and  if  the  two  solu- 
tions are  placed  in  contact,  the  reduction-process  tends  to  occur  at 
one  place  and  to  liberate  positive  electricity  there,  and  the  oxidation- 

243 


244  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

process  tends  to  occur  at  another  place  and  to  absorb  positive  electric- 
ity there,  thereby  producing  a  difference  of  potential  or  an  electro- 
motive force  between  the  two  places;  and  if  they  are  now  connected 
by  a  metallic  conductor,  a  current  of  electricity  will  flow  through  it, 
and  this  current  can  be  made  to  produce  work,  for  example,  by  passing 
it  through  an  electric  motor.  Such  an  arrangement  as  that  here 
described,  in  which  an  electromotive  force  is  produced  as  a  result  of 
an  oxidation-process  and  a  reduction-process  occurring  at  two  different 
places,  is  known  as  a  voltaic  cell. 

It  has  been  shown  in  Art.  136  that  the  quantity  of  work  that  can 
be  produced  by  any  isothermal  process  is  determined  solely  by  the 
change  in  state  of  the  system.  Hence  the  maximum  work  producible 
by  the  action  of  a  voltaic  cell  is  fully  determined  by  the  initial  and 
final  states  of  the  substances  of  which  it  consists;  and  in  any  case  under 
consideration  these  states,  or  the  corresponding  change  in  state,  must 
be  exactly  specified.  Thus,  there  must  be  stated,  in  addition  to  the 
temperature,  not  merely  the  chemical  reaction  that  takes  place  in 
the  cell,  but  also  the  conditions  of  'pressure  and  concentration  under 
which  the  substances  involved  in  it  are  produced  and  destroyed,  and 
any  transfers  of  substances  from  solutions  of  one  composition  to  those 
of  another  composition. 

In  order  that  there  may  not  be  a  finite  change  in  the  concentrations 
of  the  solutions,  and  therefore  in  the  electromotive  force  of  the  cell, 
during  the  occurrence  of  the  change  in  state,  it  will  always  be  consid- 
ered that  the  solutions  are  present  in  infinite  quantity,  so  that  when  a 
finite  quantity  of  any  substance  is  introduced  into  or  withdrawn  from 
one  of  the  solutions  of  the  cell  there  is  only  an  infinitesimal  change 
in  its  concentration.  This  method  of  consideration  is  much  simpler 
for  the  treatment  of  the  heat-content,  free  energy,  and  electro- 
motive force  of  solutes  than  the  closely  related  one,  commonly 
employed  in  the  formulations  of  the  differential  calculus,  where  the 
introduction  of  an  infinitesimal  quantity  of  the  solute  into  a  finite 
quantity  of  the  solution  would  be  considered;  for  it  enables  equiva- 
lent or  molal  quantities  of  the  solute  to  be  directly  dealt  with. 

The  character  of  the  cell  under  consideration  will  be  shown  by 
writing  the  symbols  of  the  pure  substances  and  solutions  in  the  order 
in  which  they  are  actually  in  contact  with  one  another,  commas  being 
inserted  to  indicate  the  junctions  at  which,  as  will  be  explained  later, 
an  electromotive  force  is  produced.  The  conventions  described  in 


4 


VOLTAIC  CELLS  IN  GENERAL  245 

Arts.  109  and  129  will  be  employed  to  indicate  the  state  of  aggregation 
of  substances  and  the  composition  of  solutions.  Thus  a  cell  consisting 
in  series  of  metallic  zinc,  of  a  zinc  chloride  solution  of  the  composition 
ZnCl2.iooH20,  and  of  metallic  mercury  covered  with  solid  mercurous 
chloride,  at  a  pressure  of  one  atmosphere  (as  is  always  understood 
unless  some  other  pressure  is  specified),  will  be  represented  by  the 
expression: 

Zn(s),  ZnCl2.iooH2O,  Hg2Cl2(s)  ^Hg(l). 

Similarly,  a  cell  whose  electrodes  are  an  inert  metal  M  in  contact  with 
hydrogen  gas  at  i  atm.  and  the  same  metal  in  contact  with  chlorine 
gas  at  0.05  atm.,  and  whose  electrolyte  is  a  o.i  formal  HC1  solution, 
will  be  represented  by  the  expression: 

M+H2(g)  (i  atm.),  HCl(o.i  f.),  Cl2(g)  (0.05  atm.)+M. 
Or  simply:  H2(i  atm.),  HCl(o.i  f.),  Cl2(o.o5  atm.). 

So  also  a  lead  storage-cell  whose  electrodes  are  lead  and  an  inert  metal 
coated  with  lead  dioxide,  and  whose  electrolyte  is  a  sulfuric  acid 
solution,  say  of  the  composition  H2SO4ioH2O,  saturated  with  lead 
sulfate,  will  be  represented  by  the  expression: 

Pb(s)  +PbSO4(s),  H2SO4.ioH2O,  PbSO4(s)+PbO2(s). 

/*>  The  change  in  state  taking  place  in  a  cell  may  be  expressed  by  an 
equation  whose  left-hand  member  represents  the  initial  state,  and 
whose  right-hand  member  represents  the  final  state,  of  the  substances 
involved  in  the  change.  Thus  the  changes  of  state  occurring  when  at 
25°  two  faradays  pass  from  left  to  right  (as  is  always  understood 
unless  the  opposite  is  specified)  through  the  first  two  cells  just  formu- 
lated are  shown  by  the  equations: 

Zn(s)  +  Hg2Cl2(s)  =  2Hg(l)  +Zn++Cl-2(in  ZnCl2.iooH2O)  at  25°. 
H2(i  atm.)+Cl2(o.o5  atm.)  =  2H+Cl-(at  o.i  f.)  at  25°. 

\CThe  changes  in  state  resulting  from  the  separate  electrode  processes 
/may  be  expressed  by  electrochemical  equations  similar  to  those  de- 
scribed in  Art.  63,  but  showing  also  the  composition  of  the  solutions 
in  which  the  ions  are  produced  or  destroyed.  Thus  in  the  case  of  the 
first  two  of  the  above  cells  the  changes  in  state  at  the  anode  and 
cathode  may  be  represented  by  the  equations: 

Zn(s)  +  20  =Zn++  (in  ZnCl2.iooH2O). 
Hg2Cl2(s)  =  2Hg(l)  +  2C1-  (in  ZnCl2.iooH2O)  +  20. 


246          PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

When,  as  in  these  cells,  a  gaseous  or  a  solid  non-metallic  substance 
is  in  contact  with  a  metal  electrode,  the  adjoining  solution  is  under- 
stood to  be  saturated  with  that  substance.  In  such  cells  there  are 
therefore  in  reality  two  different  solutions,  even  though  for  the  sake  of 
brevity  only  one  may  be  written  in  formulating  the  cell.  Thus  these 
cells  ought  strictly  to  be  represented  by  the  following  expressions,  in 
which  5  f .  denotes  the  formal  solubility  of  the  substance  in  the  solution 
in  contact  with  it,  thus  of  Hg2Cl2  in  a  solution  of  ZnCl2.iooH2O: 
Zn(s),  ZnCl2.iooH2O,  ZnCl2.iooH2O+Hg2Cl2(s  f.),  Hg2Cl2(s)-f-Hg(l). 

H2(i  atm.),  HCl(o.i  f.)+H2(>  f.  or  0.00076  f.),  HCl(o.i  f.)-f- 
Cl2(s  f.  or  0.0044  f.),  Cl2(o.o5  atm.). 

Pb(s)+PbSO4(s),   H2SO4.ioH2O+PbSO4(s  f.),  H2SO4.ioH2O+ 
PbSO40  f.)+Pb(SO4)2(io-8f.),  PbSO4(s)+PbO2(s). 

The  recognition  of  the  fact  that  there  are  in  such  cells  two  solutions 
of  slightly  different  composition  is  of  great  importance  in  considering 
the  mechanism  of  voltaic  action  and  in  evaluating  the  electrode-poten- 
tials discussed  in  Arts.  150-152. 

When,  as  in  the  three  cells  just  considered,  the  cell  contains  two 
solutions  of  substantially  the  same  composition  (in  the  respects  that 
in  both  solutions  the  ions  present  at  considerable  concentrations  are 
the  same,  that  the  concentrations  of  these  ions  are  substantially  the 
same,  and  that  the  solvent-medium  as  a  whole  is  substantially  the 
same),  the  ion-transferences  need  not  be  considered;  for,  though  there 
is  transference  from  the  solution  around  one  electrode  to  that  around 
the  other  electrode,  there  is  not  an  appreciable  free-energy  change, 
since  the  two  solutions  have  substantially  the  same  composition. 

Changes  of  State  in  Cells  with  a  Single  Solution.  — 
Prob.  i.  —  Formulate  an  electrochemical  equation  which  expresses 
fully  the  change  in  state  that  takes  place  at  each  electrode  when  two 
faradays  pass  through  the  hydrogen-chlorine  cell  above  mentioned; 
and  derive  from  these  equations  the  expression  for  the  change  in  state 
taking  place  in  the  cell  as  a  whole. 

Prob.  2.  —  Formulate  electrochemical  equations  expressing  the 
changes  in  state  at  the  electrodes  when  two  faradays  pass  at  20° 
through  the  lead  storage-cell  formulated  above ;  and  derive  from  them 
a  corresponding  expression  for  the  change  of  state  in  the  cell  as  a  whole. 

When  the  cell  contains  around  the  two  electrodes  solutions  of 
substantially  different  composition,  as  in  the  cells  formulated  in 
Prob.  3,  it  is  necessary  to  consider  not  only  the  processes  that  take 


VOLTAIC  CELLS  IN  GENERAL  247 

place  at  the  two  electrodes,  but  also  thejtransference  that  occurs    ')/ 
inj;he_solutipns^  for  in  such  cells  the  ion-constitutents  are  transferred 
Trom  a  solution  of  one  composition  to  a  solution  of  another  composition. 
Changes  in  State  in  Cells  with   Two  Solutions. — 
Prob.  3.  —  Write  equations  expressing  (i)  the  changes  in  state  which 
take  place  as  a  result  of  the  electrolysis,  (2)  the  changes  in  state 
brought  about  by  the  transference  of  the  ions,  and  (3)  the  resultant 
change  in  state  in  the  cell  as  a  whole,  when  one  faraday  passes  at  18° 
through  each  of  the  following  cells,  taking  0.166  as  the  transference- 
number  of  the  chloride-ion  in  the  two  HC1  solutions: 

a.  H2(i  atm.),  HCl(o.oi  f.),  HCl(o.i  f.),  H2(i  atm.). 

b.  Cl2(i  atm.),  HCl(o.oi  f.),  HCl(o.i  f.),  Cl2(i  atm.). 

Note  that  the  transference  effects  in  these  cells  are  similar  to  those 
described  in  Art.  64,  the  left-hand  solution  being  regarded  as  the  anode- 
portion  and  the  right-hand  solution  as  the  cathode-portion. 

145.  The  Production  of  Work  and  the  Corresponding  Decrease  of 
Free  Energy  in  Voltaic  Cells.  —  In  general,  when  a  quantity  of  positive 
electricity__Q_jlows  between  two  places,  such  as  the  electrodes  of  a 
voltaic  cell,  between  which  there  is  a  fall  of  potential  or  electromotive 
force jjuia  the  direction  of  the  flow,  a  quantity  of  work  equal  to  the 
product  E£-ean  be  produced.  Now,  according  to  Faraday's  law 
(Art.  62),  the  quantity  of  electricity  flowing  through  a  voltaic  cell  is 
strictly  proportional  to  the  number  of  equivalents  N  that  are  involved  in 
the  chemical  change  at  each  electrode;. that  is,^Q=NF,  where  F  repre- 
sents the  quantity  of  electricity  (96,500  coulombs)  that  passes  when  a 
reaction  involving  one  equivalent  of  each  of  the  reacting  substances 
takes  place  at  each  electrode.  The  maximum  electrical  work  that 
can  be  obtained  when  N  faradays  of  electricity  pass  through  a  voltaic 
cell  which  produces  an  electromotive  force  E  is  therefore  equal  to  E  N  F. 
When  a  voltaic  cell  acts  reversibly  there  is  ordinarily  produced, 
in  addition  to  the  electrical  work,  a  quantity  of  mechanical  work 
corresponding  to  the  changes  in  volume  of  the  different  parts  of  the 
cell  taking  place  under  their  respective  pressures.  The  total  work 
WR  is  therefore  ENF-J- S(^Az>).  Since  by  definition  (Art.  136)  the 
'free-energy  decrease  is  equal  to  WR— A(S^),  it  is  equal  simply  to  the 
electrical  work  that  can  be  produced.  That  is,  for  any  change  in 
state  taking  place  in  a  voltaic  cell  under  constant  pressures: 

—  AF=ENF. 

The  numerical  value  of  the  electromotive  force  E  of  the  cell  will 
in  this  book  be  given  a  positive  sign  when  the  cell  tends  to  produce 


248          PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

a  current  of  positive  electricity  through  the  cell  in  the  direction  in 
which  it  is  written,  and  a  negative  sign  when  the  cell  tends  to  pro- 
duce such  a  current  in  the  opposite  direction.  The  symbol  N  will  denote 
the  number  of  faradays  of  positive  electricity  that  are  considered 
to  pass  through  the  cell  from  left  to  right,  its  numerical  value  being 
given  a  negative  sign  when  positive  electricity  is  considered  to  pass 
in  the  opposite  direction.  The  maximum  electrical  work  and  the 
free-energy  decrease  calculated  by  the  equation  will  then  have,  as 
usual,  a  positive  sign  when  the  cell  produces  electrical  work,  and  a 
negative  sign  when  external  work  is  expended  upon  the  cell.  Its 
value  will  be  in  joules  when  the  electromotive  force  is  in  volts  and  the 
quantity  of  electricity  in  coulombs. 

The  Maximum  Work  and  Free-Energy  Decrease  Attending  Changes 
in  Voltaic  Cells.  — 

Prob.  4. — -The  electromotive  force  at  15°  of  the  Daniell  cell, 
Zn(s)  +ZnSO4.7H2O(s),  ZnSO4(sf.),  CuS04(s f.),  CuSO4.5H2O(s)  +Cu(s), 
is  1.093  volts.  Give  the  value  of  NF  in  coulombs  and  of  the  electrical 
work  in  joules  required  to  deposit  iZn  on  the  electrode;  and  state  what 
the  sign  of  each  of  these  quantities  signifies. 

Prob.  5.  —  a.  Calculate  exact  values  of  the  electrical  work  and  the 
mechanical  work  that  are  produced  when  i  f araday  passes  under  revers- 
ible conditions  through  the  cell  H2(i  atm.),  HCl(o.i  f.),  Cl2(o.o5  atm.) 
at  25°.  The  electromotive  force  of  this  cell  is  1.451  volts. 
The  increase  in  the  volume  of  an  infinite  quantity  of  o.i  f.  HC1  solu- 
tion caused  by  introducing  iHCl  into  it  is  18.7  ccm.  b.  Find  the 
corresponding  free-energy  decrease. 


CELLS   UNDERGOING  PHYSICAL  CHANGES  249 

II.    THE   ELECTROMOTIVE    FORCE    OF   CELLS    UNDERGOING    ONLY 
CHANGES   IN    CONCENTRATION  OR  PRESSURE 

146.  Change  of  the  Electromotive  Force  of  Voltaic  Cells  with  the 
Concentration  of  the  Solutions.  —  The  considerations  of  the  preceding 
articles  make  it  possible  to  calculate  the  change  that  is  produced  in 
the  electromotive  force  of  a  cell  by  varying  the  concentration  of  the 
solutions  contained  in  it.  Thus  the  difference  between  the  electro- 
motive forces  of  the  cells  H2(iatm.),  HCl(o.ooi  f.),  AgCl(s)+Ag(s) 
and  H2(i  atm.),  HCl(o.i  f.),  AgCl(s)+Ag(s)  can  be  derived  by  con- 
sidering that  one  faraday  passes  through  these  two  cells  arranged  in 
series  in  opposition  to  each  other,  by  noting  what  the  resultant  change 
in  state  is,  and  by  equating  the  two  expressions  for  the  attendant 
free-energy  decrease  derived  in  Arts.  145  and  138. 

The  exact  treatment  of  such  cells  is  complicated  by  the  uncertainty 
as  to  the  degree  of  ionization  of  largely  ionized  substances  and  by  the 
fact  that  their  ions  are  not  perfect  solutes.  A  rough  approximation 
to  the  truth  is,  however,  secured  in  electrochemical  calculations,  as  hi 
those  relating  to  mass-action,  by  employing  the  hypotheses  that  these 
substances  are  completely  ionized  up  to  moderate  concentrations, 
such  as  i  formal,  and  that  their  ions  are  perfect  solutes.  These 
hypotheses  are  to  be  applied  to  these  substances  in  all  the  problems 
of  this  chapter,  unless  otherwise  stated.  The  errors  to  which  these 
hypotheses  lead  will  be  indicated  in  some  cases  by  a  comparison  of  the 
calculated  with  the  observed  values. 

Effect  of  Concentration  on  Electromotive  Force.  — 

Pro}),  6.  —  Calculate  the  difference  between  the  electromotive  forces  at 
25°  of  the  two  cells  formulated  in  the  preceding  text.  Compare  the 
result  with  the  difference  between  the  observed  electromotive  forces  of 
these  cells,  which  are  0.5789  and  0.3522  volt. 

prob.  7.  —  Calculate  the  difference  between  the  electromotive  force 
at  25°  of  the  cell  H2(i  atm.),  HCl(io  f.),  Cl2(i  atm.)  and  that  of  the 
cell  H2(i  atm.),  HC1(6  f.),  Cl2(i  atm.).  The  vapor-pressures  of  HC1 
in  the  two  solutions  at  25°  are  4.2  and  0.14  mm. 

pr Ob.  s.  —  State  what  data  are  needed  in  order  to  calculate  the  electro-  M 
motive  force  EJ  at  18°  of  the  cell  H2(i  atm.),  H2SO4(io  f.),  O2(i  atm.) 
from  that  E2  of  the  cell  H2(i  atm.),  H2SO4(o.oi  f.),  O2(i  atm.);  and 
formulate  an  expression  by  which  the  calculation  could  be  made. 

Prob.  p.  -The  cell  Ag(s)  +AgCl(s),  HCl(o.i  f.),  Cl2(i  atm.)  has  at  25° 
an  electromotive  force  of  1)135  volts.  How  much  would  its  electro- 
motive force  be  changed  by  substituting  HCl(o.oi  f.)  for  the  HCl(o.i  f.)? 


250  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

*Measurements  with  such  combined  or  double  cells  form  the  basis 
of  one  of  the  important  methods  by  which  values  of  the  activity- 
coefficients,  such  as  are  given  hi  the  table  of  Art.  113,  can  be 
derived.  The  value  of  the  electromotive  force  gives  directly 
the  free-energy  decrease,  and  from  this  by  equation  (4)  of  Art.  138 
the  ratio  of  the  products  of  the  activities  of  the  ions  in  the  two  solu- 
tions .and  of  their  activity-coefficients  can  be  computed,  as  is  illus- 
trated by  the  following  problems. 

* Derivation  of  I  on- Activity-Coefficients  from  Electromotive  Forces.  — 
*Prob.  10.  —.a.  From  the  observed  values,  0.5789  and  0.3522,  for  the 
electromotive  forces  of  the  cells  formulated  in  the  text  and  considered 
in  Prob.  6,  calculate  the  ratio  of  the  products  of  the  activities  of  hydro- 
gen-ion and  chloride-ion  in  o.i  formal  and  in  o.ooi  formal  HC1  solution. 
b.  Find  the  ratio  of  the  ion-activity-coefficients  at  o.i  formal  and 
o.ooi  formal,  assuming  that  hydrogen-ion  and  chloride-ion  have  equal 
activities  in  the  same  solution. 

*Prob.  ii.  —  At  25°  0.224  volt  is  the  electromotive  force  of  the  cell: 
Hg(l)  +HgO(s) ,  KOH(o.45  f .) ,  K^l) ,  KOH(o.oo45  f  •) ,  HgO(s)  +Hg(l) , 
where  KwHg  represents  a  solution  of  potassium  in  mercury  of  definite 
but  unknown  concentration.  Find  the  ratio  of  the  ion-activity-coeffi- 
cients of  KOH  in  the  two  solutions,  assuming  that  the  two  ions  have 
equal  activities  at  the  same  concentration. 

147.  The  Electromotive  Force  of  Concentration-Cells.  — •  A  cell 
which  consists  of  two  identical  electrodes  and  of  two  solutions  con- 
taining the  same  substance  at  two  different  concentrations  is  called  a 
concentration-cell.  The  cell,  Zn(s),  ZnCl2(o.i  f.),  ZnCl2(o.oo5  f.),  Zn(s), 
is  an  example  of  a  concentration-cell,  as  are  also  the  cells  for- 
mulated in  Prob.  3,  Art.  144. 

The  electromotive  force  of  a  concentration-cell  containing  the 
solute  at  small  concentrations  can  be  calculated  by  considering  the 
changes  in  concentration  that  occur  at  each  electrode  as  a  result 
both  of  the  electrolysis  and  the  transference,  when  N  faradays  pass 
through  it,  by  formulating  in  accordance  with  Art.  138  an  expression 
for  the  free-energy  decrease  attending  these  concentration  changes, 
and  by  equating  this  expression  with  E  N  F. 

Evaluation  of  the  Electromotive  Force  of  Concentration  Cells.  — 
Prob.  12.  —  Calculate  the  electromotive  force  at  18°,  a,  of  the  cell 
formulated  in  Prob.  30  above;  b,  of  the  cell  formulated  in  Prob.  36. 
c.  Formulate  general  algebraic  expressions  involving  the  concentrations 
c'  and  c"  and  the  transference-number  TA  or  TC,  for  the  electromotive 
force  of  cells  like  these,  in  which  the  cation,  or  the  anion,  is  involved  in 
the  electrode  process.  Ans.  a,  0.0192. 


CELLS  UNDERGOING  PHYSICAL  CHANGES  251 

Prob.  13.  —  Calculate  the  electromotive  force  at  18°  of  the  following 
cell,  using  as  the  transference-number  that  derived  from  the  ion-con- 
ductances given  in  Art.  76:  Zn(s),  ZnCl2(o.i  f.),  ZnCl2(o.oo5  f.),  Zn(s). 
Ans.  —0.0660. 

*Prob.  14.  —  Calculate  the  electromotive  force  at  25°  of  the  cell 
Ag+AgCl,  KCl(o.s  f.),  KCl(o.osf.),  AgCl+Ag,  taking  0.504  as  the 
transference-number  of  the  chloride-ion,  a,  assuming  that  the  salt  is 
completely  ionized  and  that  its  ions  are  perfect  solutes;  b,  assuming  that 
the  salt  has  ionization  values  equal  to  the  conductance-viscosity  ratios 
as  given  in  the  table  in  Art.  113  and  that  its  ions  are  perfect  solutes; 
c,  using  the  values  of  the  ion-activity-coefficients  given  in  the  table  in 
Art.  113.  d.  Tabulate  these  three  results  together  with  the  experi- 
mentally determined  value  (0.0536  volt)  of  the  electromotive  force. 

Another  type  of  concentration-cell  is  that  in  which  there  is  a  single 
aqueous  solution  in  contact  with  electrodes  consisting  of  some  metal 
dissolved  at  two  different  concentrations  in  mercury.  For  example, 
the  cell  Zn.iooHg,  ZnCl2.iooH2O,  Zn.2ooHg  is  of  this  type.  Since 
most  of  the  metals  dissolved  in  mercury  at  small  concentrations  have 
been  shown  by  vapor-pressure  and  freezing-point  measurements  to  be 
nearly  perfect  solutes  having  monatomic  molecules,  the  electromotive 
force  of  cells  having  such  solutions  as  electrodes  can  be  calculated 
with  the  aid  of  the  preceding  considerations. 

A  similar  type  of  concentration-cell  is  that  in  which  the  two  elec- 
trodes consist  of  some  inert  metal  surrounded  by  the  same  gas  at 
two  different  pressures,  as  in  the  oxygen  cell  M+O2(i  atin.),  KOH(i  f.), 
02(o.iatm.)-fM. 

Electromotive  Force  of  Cells  with  the  Electrode  Substance  at  Different 
Concentrations.  — 

Prob.  15.  —  Calculate  the  electromotive  force  at  18°  of  the  zinc- 
mercury  cell  formulated  in  the  preceding  text. 

Prob.  16.  —  Calculate  the  electromotive  force  at  25°  of  the  oxygen 
cell  formulated  in  the  preceding  text. 


252  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

HI.    THE    ELECTROMOTIVE    FORCE    OF    CELLS    UNDERGOING    CHEMICAL 
CHANGES   AND   ITS   RELATION   TO   THEIR   EQUILIBRIUM  CONDITIONS 

148.  Chemical  Changes  in  Voltaic  Cells.  —  In  the  cells  thus  far 
considered  the  change  of  state  consists  only  in  a  transfer  of  one  or 
more  of  the  substances  from  one  pressure  or  concentration  to  another. 
In  most  voltaic  cells,  however,  a  chemical  change  takes  place.     Thus 
in  the  Daniellcell  the  reaction  Zn(s)+CuSO4  =  Cu(s)+ZnSO4  occurs. 
Other  examples  of  cells  in  which  similar  chemical  changes  take  place 
are  given  in  Art.  144. 

A  chemical  change  somewhat  different  in  character  from  those 
just  considered  takes  place  in  cells  whose  half-cells  consist,  not  of  solid 
or  gaseous  elementary  substances  in  contact  with  solutions  of  their 
ions,  but  of  an  inert  metal  electrode  in  contact  with  two  solutes  in 
different  stages  of  oxidation.  Thus  the  electrode-reactions  in  the  cell 
M,  Fe^Cl-2(<:i)+Fe+++a-3fe),  Cl-faJ+Ckfa),  M,  attending  the 
passage  of  one  faraday  are  Fe++-f-®  =Fe'HH-  and  f  C12  =  C1-+  0  , 
and  the  whole  reaction  is  Fe++Cl-2+iCl2  =  Fe+++Cl-3.  Such  cells  do 
not,  however,  require  special  treatment;  for  the  principles  applicable 
to  other  cells  can  be  readily  extended  to  them,  as  will  be  seen  in  the 
following  articles. 

149.  The  Electromotive  Force  of  Cells  in  Relation  to  the  Equi- 
librium Conditions  of  the  Chemical  Reactions.  —  The  free-energy 
decrease  attending  chemical  changes  between  substances  at  any 
pressures  or  concentrations  is  given  (as  shown  in  Art.  145)  by  the 
general  expression  —  AF  =  ENF,  in  which  E  represents  the  electromotive 
force  of  a  voltaic  cell  in  which  the  chemical  change  takes  place  re- 
versibly.    Another  form  of  expression  for  the  free-energy  decrease 
attending  chemical  changes  between  substances  at  small  pressures  or 
concentrations  was  derived  in  Arts.  140-142  from  a  consideration 
of  their  equilibrium  conditions.     Combination  of  these  two  expressions 
leads  to  the  following  relation  between  the  electromotive  force  E  of 
the  cell,  the  equilibrium-constant  K  of  the  chemical  change  taking 
place  in  it,  and  the  actual  concentrations,  CE',  cr',.  .CA',  CB',  in  the 
cell  of  the  chemical  substances  involved  in  the  chemical  change: 


-AF-zrt.p-.Rr 


(logtf-log 


When  a  gaseous  substance  is  involved  in  the  chemical  change,  its 
actual  pressure  may  be  substituted  for  its  concentration  in  the  last 


CELLS   UNDERGOING  CHEMICAL  CHANGES  253 

term  of  this  expression;  but  in  that  case  its  equilibrium  pressure  must 
be  correspondingly  substituted  in  the  expression  for  the  equilibrium- 
constant. 

This  expression  makes  possible  the  calculation  of  equilibrium- 
constants  from  electromotive  forces,  or  conversely  that  of  electro- 
motive forces  from  equilibrium-constants,  as  shown  in  the  following 
problems. 

Calculation  of  Equilibrium-Constants  from  Electromotive  Forces.  — 
Prob.  17.  —  Calculate  the  dissociation-pressure  in  atmospheres  of 
solid  AgCl  at  25°  from  the  electromotive  force,  1.135  volts,  of  the  voltaic 
cellAg(s)+AgCl(s),  HCl(o.i  f.),  Cl2(i  atm.).    Ans.  4.2Xio-39. 

Prob.  18. —  a.  Calculate  the  equilibrium-constant  at  25°  of  the  reac- 
tion H2(g)+2AgCl(s)=2H++2Cl-+2Ag(s)  from  the  electromotive 
force,  0.3522  volt,  of  the  cell  H2(i  atm.),  HCl(o.i  f.),  AgCl(s)+Ag(s). 
b.  Calculate  the  equilibrium  pressure  in  atmospheres  of  the  hydrogen 
gas  that  would  be  produced  by  the  action  of  metallic  silver  on 
i  f.  HC1  solution.  Ans.  6,  i.2Xio~12. 

Prob.  ig.  —  a.  From  the  data  given  in  Probs.  17  and  i8a  find  the 
electromotive  force  of  the  cell  H2(i  atm.),  H+Cl~(o.i  f.),  Cl2(i  atm.). 
b.  Calculate  from  this  electromotive  force  the  equilibrium-constant  of 
the  reaction  H2(g)  +Cl2(g)  =  2H++2C1-./  c.  Calculate  this  equilibrium- 
constant  from  the  equilibrium-constants  found  in  Probs.  17  and  i8a. 
Ans.  b.  1.9  Xio50. 

Calculations  of  Electromotive  Forces  from  Equilibrium-Constants.  — 
Prob.  20.  —  a.  Calculate  the  electromotive  force  at  25°  of  the  cell 
H2(i  atm.),  H2SO4  (o.oi  f.),  O2(i  atm.)  from  the  equilibrium-constant 
at  25°  of  the  reaction  2H20(g)  =2H2(g)+O2(g),  which  has  been  in- 
directly derived  from  equilibrium  measurements  at  higher  temperature 
and  found  to  have  the  value  9.7  Xio~81.  The  vapor-pressure  of  water 
at  25°  is  24  mm.  b.  Calculate  the  minimum  electrical  energy  re- 
quired to  decompose  iH2O  by  the  electrolysis  of  a  o.i  f.  KOH  solution 
at  25°  under  a  barometric  pressure  of  784  mm.  Ans.  &,  1.228. 

Prob.  21.  —  a.  Calculate  the  electromotive  force  at  25°  of  the  cell 
Cl2(i  atm.),  HCl(io  f.),  O2(i  atm.)  from  the  equilibrium-constant  of  the 
reaction  4HCl(g)  +  O2(g)  =  2H2O(g)  +  2Cl2(g),  which  has  been  derived  » 
from  measurements  at  higher  temperatures  of  the  equilibrium  of  this 
reaction  (involved  in  the  Deacon  process  of  producing  chlorine),  and 
found  to  have  at  25°  the  value  i.oXio13.  The  vapor-pressures  at  25° 
of  H2O  and  HC1  in  10  f.  HC1  solution  are  9.4  mm.  and  4.2  mm.,  respec- 
tively, b.  Assuming  the  process  could  be  made  reversible,  calculate 
how  much  electrical  energy  might  be  obtained  in  producing  i  mol  of 
chlorine  gas  by  an  electric  current  passing  through  a  10  f.  HC1  solution 
at  25°  between  electrodes  of  an  inert  metal  under  a  barometric  pressure 
of  750  mm.,  the  solution  around  the  cathode  being  kept  saturated  with 
air.  Ans.  0,0.1151. 


254  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

IV.    ELECTRODE-POTENTIALS  AND   LIQUID-POTENTIALS 

150.  The  Nature  of  Electrode-Potentials.  —  The  electromotive 
force  produced  by  a  voltaic  cell  is  the  sum  of  the  electromotive  forces 
produced  at  the  junctions  between  the  electrodes  and  the  solutions 
and  of  the  electromotive  forces  produced  at  the  junctions  between  the 
different  solutions  that  may  be  present  in  the  cell.  Thus  the  electro- 
motive force  of  the  Daniell  cell  is  the  algebraic  sum  of  the  electro- 
motive force  from  the  zinc  to  the  zinc  sulfate  solution,  that  from  the 
zinc  sulfate  solution  to  the  copper  sulfate  solution,  and  that  from  the 
copper  sulfate  solution  to  the  copper.  The  electromotive  forces  at  the 
electrodes  are  commonly  called  electrode-potentials;  and  those  at  the 
junctions  of  the  solutions,  liquid-potentials  (or  diffusion-potentials). 

The  consideration  of  these  partial  electromotive  forces  is  important 
in  the  respects  that  it  shows  more  clearly  the  separate  factors  which 
determine  electromotive  force  and  the  mechanism  by  which  it  is 
produced;  that  it  enables  the  electromotive  force  of  a  large  number 
of  cells  to  be  calculated  from  a  small  number  of  experimentally  deter- 
mined constants;  and  that  it  affords  a  simple  means  of  predicting  the 
equilibrium  conditions  of  oxidation  and  reduction  reactions.  The 
factors  on  which  electrode-potentials  depend  will  be  first  considered. 

Like  ordinary  chemical  changes,  the  electrochemical  reactions  that 
occur  at  electrodes  have  definite  equilibrium  conditions,  and  take 
place  in  one  direction  or  the  other  till  these  conditions  are  satisfied. 
For  example,  when  a  neutral  plate  of  copper  is  placed  in  a  o.i  formal 
CuSO4  solution  at  25°,  the  equilibrium  conditions  of  the  reaction 
Cu(s) + 2  ©  =  CU++  require  that  copper  ions  deposit  on  the  plate.  This 
gives  the  plate  a  positive  charge,  and  raises  it  to  a. higher  potential 
than  the  solution.  This  creates  a  counter  electromotive  force  which 
tends  to  drive  positive  electricity,  and  therefore  the  positively  charged 
copper  ions,  back  into  the  solution.  Equilibrium  is  established  when 
this  counter  electromotive  force  just  compensates  the  electromotive 
force  arising  from  the  mass-action  tendency  of  the  copper  ions  to 
deposit.  In  other  cases  the  tendency  of  the  reaction  to  take  place 
may  produce  an  electromotive  force  directed  from  the  metal  to  the 
solution.  For  example,  in  order  to  satisfy  the  equilibrium  conditions 
of  the  reaction  Zn(s)  + 2©  =Zn++,  zinc  placed  in  contact  with  o.i 
formal  ZnSC>4  solution  must  pass  into  the  solution  as  zinc  ions  until 
enough  positive  electricity  accumulates  in  the  layer  of  solution  adjoin- 
ing the  electrode  to  produce  a  counter  electromotive  force  equal  to  that 


ELECTRODE-POTENTIALS  AND  LIQUID-POTENTIALS     255 

corresponding  to  the  tendency  of  the  zinc  to  pass  into  solution.  In 
general,  starting  with  a  definite  concentration  of  the  ion  and  definite 
pressure  of  the  gas  (in  case  a  gas  is  involved)  at  a  definite  temperature 
and  with  electrically  neutral  conditions,  the  reaction  will  proceed 
on  the  surface  of  the  electrode  in  one  direction  or  the  other  till  enough 
positive  or  negative  electricity  is  accumulated  on  the  electrode  and 
withdrawn  from  the  adjoining  layer  of  solution  to  produce  a  potential- 
difference  sufficiently  great  to  compensate  the  inherent  tendency  of 
the  chemical  reaction  to  take  place. 

The  quantity  of  ion  that  has  to  deposit  on  the  electrode  or  be 
produced  in  the  solution  in  order  to  bring  the  electrode  to  the  equi- 
librium potential,  which  seldom  exceeds  one  or  two  volts,  is  however 
extremely  small,  owing  to  the  very  large  value  of  the  faraday  in  rela- 
tion to  the  quantity  of  electricity  which  produces  considerable  elec- 
trostatic effects.  Thus  when  a  large  glass  flask  containing  dilute 
sulfuric  acid  is  coated  on  the  outside  with  tinfoil  and  the  latter  is 
charged  with  positive  electricity  at  1000  volts,  and  when  a  capillary 
mercury  electrode  is  inserted  in  the  acid  solution  and  this  electrode  is 
connected  to  earth,  it  is  found  that  a  bubble  of  gas  consisting  of  io~10 
equivalent  of  hydrogen  is  set  free  on  the  surface  of  the  mercury. 

151.  The  Expression  of  Electrode-Potentials.  —  Although  attempts 
have  been  made  to  determine  by  various  experimental  methods 
the  absolute  values  of  electrode-potentials,  yet  the  measurements 
made  by  these  methods  are  not  nearly  so  accurate  nor  reliable  as  those 
of  the  electromotive  force  of  ordinary  cells.  It  is  therefore  customary 
to  adopt  as  the  value  of  the  electrode-potential  of  any  half-cell  (such 
as  Zn(s),ZnCl2.iooH2O)  the  electromotive  force  of  a  whole  cell  which 
consists  of  the  half-cell  under  consideration  combined  with  a  standard 
half-cell.  The  electrode-potential  of  the  standard  half-cell  is  thereby 
arbitrarily  assumed  to  be  zero.  Various  half-cells  have  been  used 
as  such  standards  of  reference;  but  it  has  now  become  a  fairly  uniform 
practice  to  employ  the  mold  hydrogen  electrode,  which  consists  of 
hydrogen  gas  at  a  partial  pressure  of  one  atmosphere  in  contact,  with 
the  aid  of  an  inert  metal  electrode  (such  as  platinized  platinum),  with 
an  aqueous  solution  i  molal  in  hydrogen-ion.  This  half-cell  is  re- 
presented by  the  formula  H2(i  atm.),  H+(i  m.).  In  this  book  this 
half-cell  will  always  be  used  as  the  standard  of  reference;  it  being 
further  specified  that  the  solution  has  such  a  hydrogen-ion  concen- 
tration that  it  produces  the  same  thermodynamic  and  mass-action 


256  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

effects  as  a  solution  containing  i  mol  of  hydrogen-ion  per  1000  grams  of 
water  would  produce  if  the  hydrogen-ion  were  a  perfect  solute  at  that 
concentration.  (*  In  other  words>  the  activity  as  denned  in  Art.  113, 
not  the  concentration,  of  the  hydrogen-ion  is  to  be  i  molal.)  The 
electrode-potential  of  any  half-cell  is  therefore  equal  to  the  electro- 
motive force  of  the  whole  cell  formed  by  combining  it  with  this  stand- 
ard half -cell.  Thus  the  electrode-potential  of  the  half -cell  Zn(s), 
ZnCl2.iooH2O  is  equal  to  the  electromotive  force  of  the  whole  cell 
Zn(s),  ZnCl2.iooH2O  ||  H+(i  m.),  H2(i  atm.);  and  the  electrode-po- 
tential of  the  half-cell  Cl2(i  atm.),  HCl(o.i  f .)  is  equal  to  the  electromo- 
tive force  of  the  cell  Cl2(i  atm.),  HCl(o.i  f.)  ||  H+(i  m.),  H2(i  atm.). 
It  is,  however,  further  understood  that  in  evaluating  this  electromotive 
force  the  liquid-potential  between  the  two  solutions  has  been  sub- 
tracted from  the  measured  electromotive  force  of  the  whole  cell  —  a 
fact  which  is  indicated  in  the  formulation  of  the  cell  by  inserting 
parallel  lines,  instead  of  a  comma,  at  the  liquid  junction,  as  is  done  in 
the  cells  just  considered. 

It  follows  from  these  conventions  that  the  electrode-potential  has  a 
positive  sign  when  positive  electricity  tends  to  flow  from  the  electrode 
to  the  solution,  and  a  negative  sign  when  it  tends  to  flow  in  the  reverse 
direction.  In  using  data  from  outside  sources,  it  is  to  be  noted  that 
foreign  electrochemists  employ  the  opposite  convention  as  to  the  sign 
of  electrode-potentials. 

In  the  case  of  liquid-potentials  the  convention  here  adopted  is  the 
same  as  in  the  case  of  whole  cells;  namely,  the  liquid-potential  is 
given  a  positive  sign  when  positive  electricity  tends  to  flow  from  the 
solution  whose  symbol  is  written  on  the  left-hand  side  to  the  solution 
whose  symbol  is  written  on  the  right-hand  side;  and  it  is  given  a 
negative  sign  in  the  reverse  case. 

Prob.  22.  —  Conventions  Relating  to  Electrode-Potentials  and  Liquid- 
Potentials.  —  a.  Formulate  the  whole  cell  whose  electromotive  force 
is  equal  to  the  electrode-potential  of  the  half-cell  Ag(s)+AgCl(s), 
KCl(o.i  f.).  b.  From  the  facts  that  at  25°  this  electrode-potential  is 
—0.289  volt  and  that  of  H2(i  atm.),  HCl(o.i  f.)  is  0.064  volt,  and  that 
the  liquid-potential  of  KCl(o.i  f.),  HCl(o.i  f.)  is  —0.028  volt,  calculate 
the  electromotive  force  of  the  cell 

H2(i  atm.),  HCl(o.i  f.),  KCl(o.i  f.),  AgCl(s)+Ag(s). 

152.  Change  of  Electrode-Potentials  with  the  Ion-Concentrations, 
and  the  Concept  of  Molal  Electrode-Potentials.  —  Just  as  the  total 


ELECTRODE-POTENTIALS  AND  LIQUID-POTENTIALS      257 

electromotive  force  of  a  cell  is  determined  solely  by  the  change  in 
state  that  takes  place  in  it,  so  any  electrode-potential  is  determined 
solely  by  the  change  in  state  that  takes  place  at  the  electrode.  With 
the  aid  of  this  principle,  by  equating  the  two  general  expressions  for 
free-energy  decrease  derived  in  Arts.  138  and  145,  the  change  of  an 
electrode-potential  with  the  concentration  of  the  ions  or  with  that  of 
any  other  gaseous  or  dissolved  substance  involved  in  the  electrode 
reaction  can  be  formulated,  as  shown  by  the  following  problems, 
provided  the  concentrations  are  so  small  that  the  dissolved  substances 
behave  as  perfect  solutes. 

Prob.  23.  —  Change  of  Electrode-Potential  with  the  Concentration  or 
Pressure  of  the  Substances  Involved  in  the  Electrode  Reaction.  —  a.  For- 
mulate an  electrochemical  equation  expressing  fully  the  change  in  state 
that  occurs  when  two  faradays  pass  from  the  electrode  to  the  solution 
in  the  half-cell  Cl2(/>  atm.),  Cl~(c  m.),  and  give  an  expression  for  the 
free-energy  decrease  in  terms  of  Its  electrode-potential  E.  b.  Describe 
a  process  of  producing  this  same  change  in  state  which  involves  three 
steps,  of  which  one  is  brought  about  with  the  aid  of  half-cell  C12  (i  atm.), 
Cl~(i  m.),  whose  electrode-potential  is  E,  and  formulate  expressions 
for  the  free-energy  decrease  that  attends  each  of  the  three  steps  of  the 
process  and  the  process  as  a  whole,  regarding  the  substances  as  perfect 
solutes,  c.  Combine  these  results,  in  accordance  with  the  law  of 
initial  and  final  states,  so  as  to  give  an  expression  for  the  difference  in 
the  electrode-potentials  of  the  two  half-cells. 

The  electrode-potential  of  a  half-cell  in  which  all  the  substances 
taking  part  in  the  electrode  reaction  are  considered  to  be  perfect 
solutes  at  a  concentration  of  i  molal  or  perfect  gases  at  a  pressure  of 
i  atm.  is  called  the  molal  electrode-potential  E  of  the  half-cell.  The 
electrode-potential  E  of  any  half-cell  in  which  the  electrode  reaction 
is  expressed  by  the  equation: 

a  A(at  CA).  .  +b B(at  p*).  .  +N0  =<?E(at  CB).  .  -f/F(at  pr).  . 

is  related  to  the  molal  electrode-potential  E  of  the  reaction  in  the  way 
shown  by  the  general  expression: 

RT.      CE*.  ./>/..      \ 
E  =  E-^10^-^VT' 

This  equation  holds  true  strictly  only  for  perfecTsolutes  and  perfect 
gases,  and  approximately  for  solutes  or  gases  only  at  moderate  con- 
centrations or  pressures;  and  correspondingly,  the  molal  electrode- 
potential  is  not  that  actually  observed  when  the  concentrations  of  the 


258          PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

solutes  are  i  molal,  but  is  the  value  calculated  by  the  equation  for 
the  concentration  i  molal  from  the  electrode-potentials  of  half-cells 
containing  the  solutes  at  much  smaller  concentrations.  (*In  other 
words,  the  molal  electrode-potential  refers  strictly  to  an  activity, 
not  to  a  concentration,  of  i  molal.)  In  the  problems  of  this  chapter, 
however,  unless  otherwise  stated,  largely  ionized  substances  are  as  an 
approximation  to  be  regarded  as  completely  ionized,  and  their  ions 
are  to  be  treated  as  perfect  solutes  up  to  a  concentration  of  i 
molal. 

For  convenience  in  numerical  calculations  it  may  be  noted  that, 
when  ordinary  logarithms  are  used,  the  coefficient  2.T>oT,RT/Nif  pre- 
ceding the  logarithmic  term  has  the  value  i.984Xio~4r/N  volt,  or 
at  25°  the  value  0.05915^  volt.  This  last  number,  approximately 
0.059  volt,  is  evidently  the  amount  by  which  the  electrode-potential 
at  25°  changes  when  the  molality  of  any  univalent  ion  involved  in  the 
electrode  reaction  is  varied  tenfold. 

Relation  between  Actual  and  Molal  Electrode-Potentials.  — 

Prob.  24.  —  Derive  by  a  consideration  like  that  employed  in  Prob.  23 
the  general  expression  given  above. 

Prob.  25.  —  Write  the  electrochemical  reaction  that  takes  place  when 
electricity  passes  from  the  electrode  to  the  solution  in  each  of  the 
following  half -cells,  where  the  molalities  of  the  solutes  are  represented 
(as  in  mass-action  expressions)  by  their  chemical  formulas  enclosed 
within  parentheses;  and  formulate  the  corresponding  expression  for  the 
relation  between  the  molal  electrode-potential  and  the  actual  electrode 
potential  at  these  molalities: 

a.  Zn(s),  Zn++  at  (Zn++)  m. 

|Cl-at(Cl-)m. 
*•  M'  {C12  at(Cl2)m. 

M    fOH-at  (OH-)m. 
'  \O2at/>0atm. 

>++  at  (Fe++)  m. 
^+++  at  (Fe+++)  m. 

e.  Hg(l)  +Hg2S04(s),  S04=at  (SO4=)  m. 
,    ,,    f  Hg2++  at  (Hg2++)  m.+Cl-  at  (Cl~)  m. 
M>  \HgCl,  at  (HgCl,)m. 
M    JMn04=  at  (Mn04=)  m. 
g'       '  \MnOr  at  (MnOr)  m. 

Prob.  26.  —  The  potential  of  a  platinum  electrode  in  an  acid  solution 
of  potassium  iodate  and  iodine  is  found  to  depend  only  on  the  concen- 


fFe 
'iFe 


ELECTRODE-POTENTIALS  AND  LIQUID-POTENTIALS     259 

trations  of  H+,  IO3~,  and  I2.  a.  Write  the  electrode-reaction  attending 
the  passage  of  electricity  from  the  electrode  to  the  solution,  b.  For- 
mulate an  expression  by  which  the  electrode-potential  at  25°  for  any 
small  concentrations  (H+),  (IO3~),  (I2)  could  be  calculated  from  the 
molal  electrode-potential. 

Evaluation  of  Molal  Electrode-Potentials.  — 

Prob.  27.  —  Calculate  an  approximate  value,  under  the  assumption  of 
complete  ionization,  of  the  molal  electrode-potential  of  Zn(s),  Zn+f-   / 
from  the  fact  that  the  cell  Zn(s),  ZnCl2  (o.oi.f.),  HC1  (0.02  f.),  H2(g)  ( 
has  at  25°  an  electromotive  force  of  0.695  v°lt,  when  the  hydrogen  gas  is 
slowly  passed  under  a  barometric  pressure  of  754  mm.  through  the  • 
HC1  solution.    The  vapor-pressure  of  water  at  25°  is  24  mm.,  and  the 
liquic^potential  in  the  cell  is  estimated  to  be    —0.030  volt.     Ans. 
/0.766J 

Prvb.  28.  —  Calculate  the  molal  electrode-potential  at  25°,  a,  of 
Ag(s)  +AgCl(s),  Cl~,  and  b,  of  Ag(s),  Ag+,  from  the  facts  that  the  cell 
H2(i  atm.),  HCl(o.i  f.),  AgCl(s)  +  Ag(s)  has  at  25°  an  electromotive 
force  of  0.3522  volt  and  the  solubility  of  silver  chloride  at  25°  is 
i.3oXio-5  formal.  Consider  the  acid  to  be  completely  ionized  and  its 
ions  to  be  perfect  solutes.  Ans.  a,  —0.234;  b,  —0.812. 

*Prob.  29.  —  Compute  more  exact  values  of  the  two  molal  electrode- 
potentials  calculated  in  Prob.  28,  taking  into  account  the  activity- 
coefficient  of  the  ions  in  the  HC1  solution,  as  given  in  the  table  of  Art. 
113.  Ans.  a,  —  0.223;  b,  —0.801. 

*Pr  ob.  30.  —  Electrode-Potential  of  the  Normal  Calomel  Electrode.  — 
Calculate  the  electrode-potential  of  Hg(l)+Hg2Cl2(s),  KCl(i  n.)  at  25° 
from  the  following  data:     The  molal  electrode-potential  of  Hg(l)  + 
Hg2Cl2(s),  Cl~  has  been  computed  to  be  —0.270  volt  by  the  method 
illustrated  by  Prob.  28;  a  i  n.  KC1  solution  is  1.033  f-  m  KC1; 
ion-activity  coefficient  of  KC1  in  this  solution  is  0.633.    Ans. 


n 


The  difference,  -0.281  volt,  between  the  electrode-potentiai-<#  the 
normal  calomel  electrode,  which  is  the  half-cell  represented  by  the 
formula  Hg(l)  +  Hg2Cl2(s),  KCl(i  n.),  and  that  of  the  molal  hydrogen 
electrode  is  a  quantity  of  much  practical  importance,  since  other 
electrode-potentials  are  often  measured  directly  against  the  normal 
calomel  electrode,  instead  of  against  a  hydrogen  electrode,  because  of 
greater  experimental  convenience  and  because  the  liquid-potentials 
involved  are  usually  much  smaller. 

153.  Values  of  the  Molal  Electrode-Potentials.  —  The  following 
table  contains  some  of  the  more  accurately  determined  values  of  the 
molal  electrode-potentials  at  25°  and  one  atmosphere,  referred  to  that 
of  the  molal  hydrogen  electrode  taken  as  zero. 


260 


PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 


MOLAL  ELECTRODE-POTENTIALS  AT  25°. 


Reduced 

Oxidized 

Electrode- 

Reduced 

Oxidized 

Electrode- 

state 

state 

potential 

state 

state 

potential 

Li 

Li+ 

2.958 

Cu-hCl- 

CuCl(s) 

—O.II9 

Rb 

Rb+ 

2.924 

Ag+Cl- 

AgCl(s) 

—0.224 

K 

K+ 

2.922 

Hg+Cl- 

^Hg2Cl2(s) 

—O.27O 

Na 

Na+ 

2.713 

Cu 

Cu"^ 

-0-345 

Zn 

Zn++ 

0.758 

2OH- 

^02  +H2O 

-0.399 

Fe 

Fe++ 

0.441 

Cu 

Cu+ 

-0-47 

Cd 

Cd++ 

0.398 

I- 

^I2(s) 

-0.536 

Tl 

T1+ 

0.336 

Fe++ 

Fe+++ 

-0.747 

Sn 

Sn++ 

0.136 

Hg(l) 

^Hg^ 

-0.799 

Pb 

Pb++ 

O.I22 

Ag 

Ag+ 

—O.SOO 

H2(g) 

2H+ 

O.OOO 

Br~ 

£Br2(i  m.) 

-1.085 

ci- 

^Cl2(g) 

-1-359 

154.  Derivation  of  Related  Molal  Electrode-Potentials  from  One 
Another.  —  Molal  electrode-potentials  that  are  related  to  one  another 
can  be  calculated  in  the  ways  illustrated  by  the  following  problems. 

Prob.  31.  —  Molal  Electrode-Potentials  Corresponding  to  Different 
States  of  Aggregation.  —  Calculate  the  molal  electrode-potential  at  25° 
a,  of  C12  (i  m.),  Cl~~,  and  b,  of  I2  (i  m.),  I~~,  from  the  values  given  for 
Cl2(g),  Cl~,  and  for  I2(s),  I~,  in  the  table  of  Art.  153,  and  from  the 
molalities,  0.062  and  0.0013,  of  C12  and  I2  in  water  saturated  with  them 
at  25°  when  the  pressure  is  i  atm.  Ans.  a,  —1.395;  &>  —0.621. 

Prob.  32.  —  Molal  Electrode-Potential  of  a  Metal  in  the  Presence  of 
One  of  Its  Slightly  Soluble  Salts.  —  Calculate  the  molal  electrode- 
potential  at  25°  of  Hg(l)  +Hg2SO4(s),  SO4=,  from  that  of  Hg(l),  Hg2++ 
given  in  the  table,  and  from  the  solubility-  product  (Hg2++)  X  (SO4~) 
of  Hg2SO4  which  at  25°  is  8.2  X  lo"7.  Ans.  -0.62  volt. 

Prob,  33.  —  Molal  Electrode-Potentials  of  an  Element  that  Exists  in 
more  than  Two  States  of  Oxidation.  —  a.  By  considering  the  changes  in 
state  that  occur  on  passing  electricity  at  25°  through  the  voltaic  cells 


Cu,Cu+(i  m. 


m.),Cu,  and  Cu, 


m.) 


derive  a  relation  between  the  molal  electrode-potentials  of  Cu(s),  Cu+; 
Cu(s),  Cu++;  and  Cu+,  Cu++.  b.  Calculate  the  last  of  these  potentials 
from  the  first  two.  Ans.  b,  —0.22. 

155.  The  Electromotive  Force  of  Cells  with  Dilute  Solutions 
in  Relation  to  the  Molal  Electrode-Potentials.  —  The  electro- 
motive force  of  cells  in  which  the  liquid-potentials  are  negligible  or  for 
theoretical  purposes  are  to  be  disregarded  can  be  calculated  from  the 
molal  electrode-potentials  in  the  way  illustrated  by  the  following 
problems. 


ELECTRODE-POTENTIALS  AND  LIQUID-POTENTIALS    261 

Evaluation  of  Electromotive  Forces.  — 

Prob.  34.  —  Calculate  the  electromotive  force  at  25°  of  the  cell 
Zn(s),  ZnCl2(o.ooi  f.),  AgCl(s)+Ag(s). 

Prob.  35.  —  Calculate  the  electromotive  force  at  25°  of  the  cell 
H2(o.i  atm.),  HCl(o.i  f.),  PbCl2(s)+Pb(s).  The  saturated  solution 
of  PbCl2  in  water  at  25°  is  0.039  formal. 

Prob.  36.  —  Formulate  numerical  expressions  for  calculating  the 
electromotive  force  at  25°  of  each  of  the  following  cells: 
a,  M+CuCl(s),  CuCl2(o.oi  f.),  Cl2(o.i  atm.). 


The  solubility-product  of  CuCl  is  i.2Xio~6  formal.    For  the  composi- 
tion of  iodide  solutions  saturated  with  iodine  see  Prob.  45,  Art.  in. 

156.   The  Electromotive  Force  of  Cells  with  Concentrated  Solutions. 

—  The  electromotive  force  of  any  cell  in  which  solutes  are  present  at 
large  concentrations  cannot  be  calculated  from  the  molal  electrode- 
potentials  with  the  aid  of  the  logarithmic  concentration  formula  of 
'Art.  152,  since  this  holds  true  even  approximately  only  when  the 
concentration  does  not  exceed  i  formal.  A  knowledge  of  the  change 
in  state  taking  place  in  such  a  cell  and  of  the  reactions  occurring  at  its 
electrodes  is,  however,  of  importance,  since  it  shows  qualitatively 
the  factors  which  determine  the  magnitude  of  the  electromotive  force. 
This  is  illustrated  by  the  following  problems,  which  relate  to  certain 
cells  of  technical  importance. 

Prob.  37.  —  The  "Dry  Cell."  —  The  Leclanche  cell  and  the  common 
"dry  cell,"  which  is  a  Leclanche  cell  to  which  some  porous  material, 
such  as  paper-pulp  or  sawdust,  has  been  added,  consists  essentially  of  a 
zinc  rod  (amalgamated  to  diminish  local  voltaic  action)  dipping  into  a 
concentrated  NH4C1  solution  containing  ZnCl2,  and  a  carbon  rod  coated 
with  MnO2  dipping  into  the  same  solution,  a.  Formulate  the  cell  and 
the  reaction  that  takes  place  at  each  electrode  and  the  whole  reaction 
in  the  cell,  taking  into  account  the  facts  that  the  MnO2  is  reduced  to 
Mn2O3,  and  that  Zn(OH)2  is  soluble  in  NH4C1  solution  with  formation 
of  Zn(NH3)4Cl2.  b.  Show  in  what  direction  the  electromotive  force 
would  be  changed  by  decreasing  the  concentration  of  the  NH4C1. 

Prob.  38.  —  The  Edison  Storage  Cell.  —  In  the  nickel-iron  (Edison) 
storage  cell,  Fe(s)+Fe(OH)2(s),  KOH(2i%),Ni(OH)2(s)+Ni(OH)3(s), 
the  main  reaction  is  Fe(s)+2Ni(OH)3(s)  =  Fe(OH)2(s)+2Ni(OH)2(s) 
(the  degree  of  hydration  of  the  three  oxides  being,  however,  somewhat 
indefinite),  a.  Write  the  reaction  occurring  at  each  electrode,  b.  Show 
in  what  direction  each  of  the  electrode-potentials,  and  also  the 
electromotive  force  of  the  whole  cell,  would  vary  with  increase  of  the 
KOH  concentration. 


262  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

*Prob.  39.  —  The  Clark  Standard  Cell.  —  One  form  of  the  Clark  cell, 
used  as  a  standard  cell  in  electromotive  force  measurements  is  at  20° 
represented  by  the  formula:  Zn(s)+ZnSO4.7H2O(s),  ZnSO4.i6.8H2O, 
Hg2SO4(s)+Hg(l).  a.  Specify  the  change  of  state  that  occurs  when 
two  faradays  pass  through  the  cell.  b.  Show  that  even  in  an  actual 
cell,  with  only  a  finite  quantity  of  solution,  no  variation  of  the  electro- 
motive force  results  when  two  faradays  pass,  provided  the  change  in 
state  takes  place  slowly  enough,  c.  Explain  how  the  electromotive 
force  of  the  cell  would  be  changed  by  making  the  saturated  ZnSO4 
solution  also  i  formal  in  Na2SO4. 


157.  The  Nature  of  Liquid-Potentials  and  Expressions  for  their 
Approximate  Evaluation.  —  The  electromotive  force  at  the  boundary 
between  two  solutions  arises  from  the  different  rates  at  which  the 
positive  and  negative  ions  tend  to  diffuse  from  one  solution  into  the 
other. 

This  principle  may  be  first  illustrated  by  applying  it  to  solutions 
of  the  same  substance  at  two  different  concentrations,  such  as  HC1 
(i  f.),  HCl(o.i  f.).  In  this  case  both  kinds  of  ions  diffuse  from  the 
first  to  the  second  solution  in  consequence  of  their  smaller  concen- 
trations in  the  latter  solution;  but,  owing  to  their  greater  mobility, 
the  hydrogen  ions  tend  to  pass  into  the  second  solution  in  larger 
quantity,  and  they  do  so  to  a  slight  extent,  until  such  an  excess  of 
positive  electricity  is  accumulated  in  the  intermediate  layers  that  the 
unequal  diffusion-rates  of  the  two  ions  are  compensated  by  the  effect 
of  the  potential-gradient  in  driving  the  positive  hydrogen  ions  back- 
ward and  drawing  the  negative  chloride  ions  forward.  The  result 
of  the  different  diffusion-rates  of  the  two  ions  is  in  this  case  evidently 
an  electromotive  force  directed  from  the  more  concentrated  to  the 
more  dilute  HC1  solution. 

Similar  considerations  apply  to  solutions  of  two  different  sub- 
stances at  the  same  concentration,  such  as  HCl(i  f.),  NaCl(i  f.).  In 
this  case,  the  hydrogen  ions  evidently  tend  to  diffuse  from  the  first 
solution  to  the  second,  and  the  sodium  ions  in  the  reverse  direction; 
but  owing  to  the  much  greater  mobility  of  the  hydrogen  ions  there  is  a 
resultant  flow  of  positive  electricity,  and  correspondingly  an  electro- 
motive force,  from  the  HC1  to  the  NaCl  solution.  As  before,  there 
can  be  only  a  very  slight  accumulation  of  hydrogen  ions  in  the  bound- 
ary of  the  second  solution  because  of  the  counter  electromotive  force 
thereby  produced. 


ELECTRODE-POTENTIALS  AND  LIQUID-POTENTIALS      263 

Prob.  40.  —  The  Nature  and  Direction  of  Liquid-  Potentials.  —  a.  Ex- 
plain fully,  in  accordance  with  the  above  statements,  the  mechan- 
ism of  the  process  by  which  a  liquid-potential  results  in  the  case 
of  KOH(i  f.),  KCl(i  f.).  b.  Predict  the  sign  of  the  liquid-potential 
of  NaOH(i  f.),  NaOH(o.i  f.);  and  of  NaCl(i  f.),  KNO3(i  f.). 

Expressions  by  which  liquid-potentials  can  be  approximately 
evaluated  can  be  derived  either  by  formulating  quantitatively  the 
above  considerations  with  the  aid  of  the  laws  of  diffusion,  or  by  apply- 
ing the  free-energy  principles  in  the  way  shown  in  Art.  158  to  the 
changes  in  state  taking  place  at  the  boundary. 

By  these  methods  the  following  expressions  for  the  liquid-potential 
EL,  in  terms  of  the  ion-conductances,  AC,  AC',  AA,  etc.,  and  the  number 
of  charges  or  valences,  VA  and  vc,  of  the  cation  and  anion,  have  been 
derived  under  the  assumptions  that  the  substances  are  completely 
ionized  and  that  their  ions  are  perfect  solutes  and  have  constant 
mobilities  equal  to  those  at  zero  concentration:  — 

For  combinations  of  the  ionic  types  C+A~(at  ci),  C+A~(atc2); 
C+2A=(at  ci),  C+2A=(at  c2);  etc: 

A_C_AA 

F   ._  ^  _  »v   RT        Cl  (i) 

EL~  AC+AA     F    10g^ 

For  combinations  of  the  types  CA,  C'A;  C2A,  C'2A;  CA2,  C'A2; 
etc.,  in  which  the  two  substances  are  at  the  same  concentrations: 

RT.        Ac+AA 


For  combinations  of  the  types  CA,  C'A;   C2A,  C'2A;  CA2,  CA'2; 
etc.,  in  which  the  two  substances  have  the  same  concentrations: 

RT,        Ac+AA  . 


Evaluation  of  Liquid  Potentials.  — 

Prob.  41.  —  Calculate  the  liquid-potential  at  18°  of  the  combinations: 
a,  NaCl(o.i.f.),  NaCl(o.oi  f.);  &,  K2S04(o.i  f.),  K2SO4(o.oi  f.). 

Prob.  42.  —  Calculate  the  liquid-potential  at  25°  of:  o,  KOH(o.i  f.), 
KCl(o.i  f.)  ;  6,  HCl(o.oi  f.),  KCl(o.oi  f.)  ;  c,  ZnSO4(o.i  f.),  CuSO4(o.i  f.). 

Prob.  43.  —  Calculate  the  electromotive  force  at  25°  of  the  cell  Ag(s), 
AgN03(c.i  f.),  KN03(o.i  f.),  KCl(o.i  f.),  KCl(i  f.)  +Hg2Cl2(s), 


For  combinations  of  solutions  of  two  salts  having  a  common  ion 
but  different  concentrations,  such  as  KNO3(o.i  f.),  KCl(i  f.),  and 


264  PRODUCTION  OF  WORK  IN   VOLTAIC  CELLS 

for  combinations  of  solutions  of  salts  without  a  common-ion,  such  as 
KNOs(o.i  n.),  NaCl(o.oi  n.),  the  calculation  of  the  liquid-potential 
is  more  complicated.  Combinations  of  these  kinds  can,  however,  be 
avoided,  as  illustrated  by  the  cell  of  Prob.  43,  by  connecting  the  two 
solutions  through  intermediate  ones  so  as  to  produce  only  combina- 
tions of  the  two  types  for  which  the  liquid-potentials  can  be  calculated 
by  the  above  equations. 

A  liquid-potential  can  sometimes  be  experimentally  determined  by 
measuring  the  electromotive  force  of  a  cell  involving  it  in  which  the 
two  electrode-potentials  are  made  substantially  equal,  as  in  the  cells: 

Hg(l)+Hg2Cl2(s),  KCl(o.i  f.),  NaCl(o.i  f.),  Hg2Cl2(s)-f-Hg(l). 

Ag(s),  AgNO3(o.ooi  f.)+KNO8(o.i  f.),  AgNO3(o.ooi  f.)  + 
NaN03(o.i  f.),  Ag(s). 

Prob.  44.  —  a.  Show  that  the  electromotive  force  of  the  silver  cell  just 
formulated  is  substantially  equal  to  the  liquid-potential  of  KNO3(o.i  f.), 
NaNO3(o.i  f.),  by  specifying  exactly  what  determines  each  of  the  three 
partial  potentials  of  the  cell,  b.  State  what  assumption  is  involved 
in  regarding  the  two  electrode-potentials  equal  to  each  other. 

*158.  Derivation  of  the  Liquid-Potential  Equations.  —  Just  as 
the  electromotive  force  of  a  whole  cell  is  determined  by  the  free-energy 
decrease  attending  the  change  in  state  that  takes  place  in  the  cell,  and 
just  as  the  electrode-potential  is  determined  by  the  free-energy  decrease 
attending  the  change  in  state  that  takes  place  at  the  electrode,  so  a 
liquid-potential  is  determined  by  the  free-energy  decrease  attending 
the  change  in  state  that  takes  place  at  the  boundary  between  the  two 
solutions. 

The  change  in  state  at  such  a  boundary  is  of  a  simple  character 
in  the  case  where  the  two  solutions  contain  only  the  same  solute  at 
two  different  concentrations,  as  in  the  combination,  HCl(o.oi  f.), 
HCl(o.i  f.).  In  this  case,  in  order  to  derive  equation  (i)  of  Art.  157 
for  the  liquid-potential,  it  is  only  necessary  to  consider  the  number  of 
equivalents  of  the  positive  ion-constituent  and  of  the  negative  ion- 
constituent  that  pass  through  the  boundary  in  the  two  opposite 
directions  per  faraday  passed  through,  to  formulate  the  free-energy 
decrease  attending  these  transfers,  on  the  one  hand,  in  terms  of  the 
electrical  work  E^  involved  when  they  are  brought  about  by  the 
passage  of  one  faraday,  and  on  the  other  hand,  in  terms  of  the  usual 
logarithmic  concentration  equations  of  Art.  138,  and  to  equate  these 
two  expressions.  This  derivation  involves  the  assumptions  that  the 


ELECTRODE-POTENTIALS  AND  LIQUID-POTENTIALS     265 

solute  is  completely  ionized,  that  its  ions  behave  as  perfect  solutes, 
and  that  the  ratio  of  their  mobilities,  that  is,  the  transference-number, 
is  constant  up  to  the  highest  concentration  involved. 

Derivation  of  the  Equation  for  the  Liquid-Potential  between  Solutions 
with  the  Same  Solute  at  Different  Concentrations.  — 

Prob.  45.  —  Derive  equation  (i)  of  Art.  157  in  the  form  it  assumes  for  a 
uniunivalent  solute  at  two  different  concentrations,  such  as  HCl(at  Ci)  , 
HCl(at  c2),  making  the  assumptions  stated  in  the  text. 

Prob.  46.  —  Derive  equation  (i)  of  Art.  157  in  the  form  it  assumes  for 
a  unibivalent  solute  at  two  different  concentrations,  such  as  CaCl2(at  Ci), 
CaCl2(at  c2),  making  the  assumptions  stated  in  the  text. 

It  will  be  evident  from  these  derivations  that,  provided  only  that  the 
ratio  of  the  two  ion-conductances,  and  hence  the  transference-number 
TC  or  TA,  does  not  vary  between  the  two  concentrations  involved,  and 
that  the  two  ions  can  be  considered  to  have  the  same  activity-co- 
efficient a.  in  the  same  solution,  an  exact  expression  for  the  liquid- 
potential  is  given  in  the  case  of  a  uniunivalent  solute  by  the  equation: 


By  a  similar  consideration  of  the  ion-transference  that  takes 
place  at  the  boundary  of  the  two  solutions  and  with  the  aid  of  the 
same  assumptions,  an  expression  can  be  derived  for  the  liquid-potential 
of  combinations  of  the  types-KCl,  HC1;  K2SO4,  Na2SO4;ZnSO4,  CuSO4. 
Namely,  taking  as  a  specific  example  the  combination  KCl(at  c)t 
HCl(at  c),  it  will  be  noted  that  there  must  be  at  the  boundary  an 
intermediate  portion  of  the  solution  in  which  the  two  substances  are 
present  in  varying  proportions,  the  concentration  of  the  potassium 
chloride  decreasing  continuously  from  left  to  right  from  c  to  o,  and 
that  of  the  hydrogen  chloride  increasing  continuously  in  the  same 
direction  from  o  to  c.  Considering  now  within  the  boundary-portion 
any  two  adjoining  layers  of  infinitesimal  length  in  which  at  equilibrium 
the  prevailing  concentrations  of  the  two  cation-constituents  are 
CK  or  CR  and  cK+dcK  or  cH+dcQ  and  their  transference-numbers  are 
TK  and  TH,  it  is  evident,  when  one  faraday  passes  from  the  KC1  solu- 
tion to  the  HC1  solution,  that  TK  mols  of  potassium-ion  are  transferred 
from  the  concentration  CK  to  the  concentration  cK-\-dcK  and  TH  mols 
of  hydrogen-ion  from  CR  to  cH+dcs.  At  the  same  time  TCL  equivalents 
of  chloride-ion  are  transferred  in  the  opposite  direction;  but  this  need 
not  be  considered,  since  its  concentration  is  uniform  throughout  the 


266  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

whole  solution.    The  free  energy  decrease  —  dF  attending  this  transfer- 
ence is  therefore  given  by  the  equation: 


A  corresponding  transference,  attended  by  a  corresponding  free-energy 
decrease,  takes  place  between  each  pair  of  adjoining  infinitesimal  layers 
throughout  the  whole  boundary-portion.  The  total  free-energy 
decrease,  which  is  equal  to  the  electrical  work  ELE  attending  the 
passage  of  one  f  araday,  is  therefore  equal  to  the  integral  of  this  expres- 
sion, taken  for  the  CK  term  between  the  limits  CK  =  C  and  CK  =  O  and 
for  the  CR  term  between  the  corresponding  limits  cs  =  o  and  CH  =  C. 
That  is: 


Now  the  transference-number  of  any  ion-constituent,  as  indicated  in 
Art.  73,  is  equal  to  the  ratio  of  the  product  of  its  concentration  by  its 
conductance  to  the  sum  of  the  corresponding  products  for  all  the  ion- 
constituents  present.  Hence,  assuming  complete  or  equal  ionization 
for  the  two  substances  in  all  the  solutions  and  constant  ion-mobilities 
of  the  three  ions  up  to  the  concentration  c,  the  following  expressions 
result: 


Replacing  cu  in  the  first  of  these  expressions  by  c—  CK,  and  CK  in  the 
second  of  these  expressions  by  c—  CH,  substituting  the  resulting  values 
of  TK  and  TH  in  the  above  free-energy  equation,  carrying  out  the  indi- 
cated integration,  and  simplifying  the  result  by  appropriate  transfor- 
mations, there  is  obtained  the  following  equation,  corresponding  to 
equation  (2)  of  Art.  157. 


Pr  ob.  47.  —  Derivation  of  the  Equation  for  the  Liquid-Potential  between 
Solutions  with  Two  Solutes  at  the  Same  Concentrations,  —  Derive  the 
equation  just  given  by  the  method  stated  in  the  text. 

159.  Determination  of  Ion-Concentrations  and  of  Equilibrium- 
Constants  by  Means  of  Electromotive  Force  Measurements.  — 
Measurements  of  electromotive  force,  interpreted  with  respect  to  the 
electrode-potentials  involved,  furnish  an  important  means  of  deter- 


ELECTRODE-POTENTIALS  AND  LIQUID-POTENTIALS     267 

mining  ion-concentrations  which  are  so  small  that  they  cannot  be 
readily  measured  by  other  methods.  The  so  determined  ion-concen- 
trations may  be  used  for  calculating  the  solubilities  of  very  slightly 
soluble  salts  or  the  ionization-constants  or  other  equilibrium-constants 
of  chemical  reactions,  as  shown  in  the  following  problems.  Measure- 
ments of  electromotive  force  similarly  interpreted  are  frequently 
employed  by  physiologists  for  determining  the  hydrogen-ion  concen- 
tration in  blood  and  other  animal  fluids,  and  sometimes  by  chemical 
analysts  for  the  titration  of  acids  and  bases,  especially  in  cases  where 
the  color  of  the  solution  makes  the  use  of  indicators  unreliable. 

Prob.  48.  —  Determination  of  Solubility.  —  The  electromotive  force  of 
the  cell  Ag(s)+AgI(s),  KI(o.i  f.),  KNO3(o.i  f.),  AgNO,(o.i  f.),  Ag(s) 
is  0.814  volt  at  25°.  Calculate  the  solubility  of  silver  iodide  in  water 
at  25°. 

Prob.  49.  —  Determination  of  the  lonization-Constant  of  Water.  — 
Calculate  the  ionization-constant  of  water  (Art.  104)  at  18°  from  the 
fact  that  the  electromotive  force  of  the  cell  H2(i  atm.),  HCl(o.i  f.), 
KCl(o.i  f.),  KOH(o.i  f.),  H2(i  atm.)  is  -0.653  volt  at  18°. 

Prob.  50.  —  Determination  of  Complex-Constants.  —  The  electro- 
motive force  of  the  cell  Ag(s),  K+Ag(CN)r(o.oi  n.)+KCN(i  n.), 
KCl(i  n.),  Hg2Cl2(s)  +Hg(l)  is  oj^  volt  at  25°.  Calculate  the  dissocia- 
tion-constant  of  the  complex-ion  Ag(CN)2~  at  25°.  Neglect  the  liquid- 
potential,  which  is  small  in  this  case.  Ans.  i.5Xio~21. 

Prob.  51.  —  Determination  of  the  Hydrolysis  of  Salts  and  the  lonization- 
Constants  of  Slightly  Ionized  Acids.  — When  a  solution  0.05  formal  in 
Na2HPO4  is  saturated  with  hydrogen  at  i  atm.,  when  a  platinum  elec- 
trode is  placed  in  it,  and  when  the  half-cell  HCl(o.oi  f.)+NaCl(o.i  f.), 
H2(i  atm.),  is  brought  into  contact  with  it,  the  cell  thus  formed 
is  found  to  have  at  18°  an  electromotive  force  of  0.398  volt.  Calculate 
the  hydrolysis  of  the  salt  and  the  ionization-constant  of  HPO4~~,  assum- 
ing complete  ionization  of  the  largely  ionized  substances,  and  neglecting 
the  liquid-potential,  which  is  made  small  by  the  addition  of  the  sodium 
chloride. 

Prob.  52.  —  Determination  of  Indicator-Constants.  —  When  the 
Na2HPO4  solution  of  Prob.  51  is  made  o.oooi  formal  in  phenolphthalein 
the  indicator  is  found  to  show  a  color  13%  as  intense  as  that  which  is 
produced  on  adding  to  the  solution  an  excess  of  sodium  hydroxide. 
Calculate  the  ionization-constant  of  this  indicator. 

Prob.  53.  —  Electrometric  Titration  of  Acids  and  Bases.  —  Into  a 
solution  of  an  acid  to  be  titrated  are  introduced  the  side-arm  of  a 
calomel  half-cell  and  a  tube  carrying  a  small  platinized  platinum 
electrode,  around  which  hydrogen  gas  at  the  barometric  pressure  is  kept 
bubbling.  A  standard  0.5  n.  NaOH  solution  is  added  from  a  burette 
until  the  electromotive  force  between  the  hydrogen  electrode  and  the 


268  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

calomel  electrode  is  found  to  change  very  rapidly  as  more  alkali  is 
added.  Calculate  the  successive  changes  that  would  be  observed  in  the 
electromotive  force,  when  the  alkali  added  is  equivalent  to  99.0,  99.5, 
99.8,  and  100.2%  of  the  acid  present,  a,  in  case  the  acid  to  be  titrated 
is  a  o.i  normal  solution  of  HC1;  b,  in  case  it  is  a  o.i  normal  solution 
of  an  acid  of  ionization-constant  of  io~5,  referring  to  the  table  of 
Art.  1 08  for  the  hydrogen-ion  concentrations.  Neglect  the  liquid- 
potential  and  the  change  in  the  volume  of  the  solution  produced  by  the 
addition  of  the  standard  NaOH  solution. 


EQUILIBRIUM  OF  OXIDATION  REACTIONS  269 

V.    THE  EQUILIBRIUM  OF  OXIDATION  REACTIONS  IN  RELATION  TO  THE 
ELECTRODE-POTENTIALS 

160.  Derivation  of  the  Equilibrium-Constants  of  Oxidation  Re- 
actions from  the  Molal  Electrode-Potentials.  —  When  the  concentra- 
tions of  the  substances  involved  in  the  two  electrode  reactions  of  any 
cell  are  such  that  the  two  electrode-potentials  are  equal  to  each  other, 
there  is  evidently  no  tendency  for  the  cell  to  act  nor  for  the  chemical 
change  to  take  place  in  it.  In  other  words,  the  concentrations  that 
make  the  two  electrode-potentials  equal  are  concentrations  at  which 
the  chemical  change  is  in  equilibrium.  The  equilibrium-constant  of  a 
chemical  change  involving  solutes  or  gases  at  small  concentrations  or 
pressures  can  therefore  be  evaluated,  as  in  the  following  problems,  by 
conceiving  a  cell  in  which  the  reaction  would  take  place  and  calculat- 
ing from  the  molal  electrode-potentials  by  the  usual  logarithmic 
equation  the  concentrations  of  the  substances  involved  hi  the  electrode 
reaction  that  will  make  the  two  actual  electrode-potentials  equal. 

Evaluation  of  Equilibrium-Constants  by  a  Consideration  of  the  Equi- 
librium Conditions  of  Voltaic  Cells.  — 

Prob.  54.  —  a.  Calculate  the  concentration  of  copper-ion  at  which  the 
reaction  Zn(s)+Cu"H"  =  Cu(s)+Zn"H"  is  in  equilibrium  at  25°  when  the 
zinc-ion  is  i  molal.  b.  Calculate  the  equilibrium-constant  at  25°. 

Prob.  55.  —  a.  Calculate  the  concentration  of  hydrogen-ion  at  which 
the  reaction  Pb(s)+2H+Cl~  =  H2(g) +Pb'H-Cl-2  is  in  equilibrium  at 
25°  when  the  hydrogen  has  a  pressure  of  i  atm.  and  the  lead-ion  is  0.03 
molal.  b.  Formulate  an  algebraic  relation  between  the  equilibrium- 
constant  of  the  corresponding  ion  reaction  and  the  molal  electrode- 
potentials,  c.  Calculate  the  value  of  this  equilibrium-constant. 

The  foregoing  considerations  evidently  lead  to  a  general  expression 
for  the  equilibrium-constants  of  oxidation  reactions.  Namely,  the 
equilibrium-constant  K  of  the  reaction  which  takes  place  when  N 
faradays  of  electricity  pass  from  left  to  right  through  a  cell  whose 
left-hand  and  right-hand  molal  electrode-potentials  are  E\  and  E2, 
respectively,  is  related  to  these  potentials  as  follows: 
l?riog^  =  (E1-E2)  NF. 

This  relation  may  be  directly  derived  from  free-energy  considera- 
tions in  the  following  way.  The  free-energy  decrease  attending  a 
chemical  reaction  when  the  reacting  substances  and  reaction-products 
are  all  at  i  molal  (or  if  gaseous,  at  i  atmosphere)  has  been  shown  in 
Arts.  140  and  141  to  be  given  by  the  expression: 


270  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

The  free-energy  decrease  attending  this  same  change  in  state,  when 
this  is  considered  to  take  place  in  a  cell  in  which  all  the  substances 
are  at  i  molal  (or  if  gaseous,  at  i  atmosphere),  is  evidently  expressed  by 
the  equation:  ,~  ~ , 

-AF  =  NF  (Ej-Eg). 

By  equating  these  two  expressions  for  the  free-energy  decrease  the 
general  relation  given  above  obviously  results. 

It  will  be  seen  from  these  considerations  that  electrode-potentials 
are  of  great  importance  in  their  purely  chemical  relations.  When 
they  are  so  considered  they  are  appropriately  called  the  reduction- 
potentials  of  the  corresponding  oxidation  reactions.  Thus,  the,  elec- 
trode-potential (0.122  volt)  of  Pb(s),  Pb++  is  the  reduction-potential 
of  the  reaction  Pb(s)  +  28  =  Pb+  +  . 

Expressed  in  chemical  terminology,  the  preceding  equation  shows 
how  the  equilibrium-constant  K  of  a  chemical  reaction  which  is  the 
resultant  of  an  oxidation  reaction  (such  as  Pb(s)  +  20  =Pb++)  and 
of  a  reduction  reaction  (such  as  2H+  =H2(g)  +  20)  is  related  to  their 
molal  reduction-potentials,  EI  and  £2;  the  symbol  N  denoting  now  the 
number  of  positive  electrons  involved  in  the  separate  reactions. 

Evaluation  of  Equilibrium-Constants  by  the  General  Equation.  — 

Prob.  56.  —  a.  Derive  an  expression  for  the  equilibrium-constant  of 
the  reaction  Cu(s)  -f-Cu"^  =  2Cu+  in  terms  of  the  molal  reduction-poten- 
tials, b.  Calculate  the  concentration  of  cuprous  salt  resulting  when 
copper  is  shaken  with  a  o.i  f.  CuSO4  solution  at  25°.  c.  Calculate  the 
concentration  of  cupric  salt  in  the  equilibrium-mixture  produced  by 
shaking  a  c.i  f.  CuCl2  solution  with  copper  at  25°,  noting  that  solid 
CuCl  separates,  whose  solubility-product  at  25°  is  i.2Xio~6  molal. 
Ans.  b,  0.0024;  c,  0.00172. 

Prob.  57.  —  a.  Formulate  an  algebraic  expression  showing  how  the 
equilibrium -constant  of  the  reaction  Ag(s)+Fe+++=Ag++Fe~H~,  ex- 
pressed in  terms  of  the  equilibrium  concentrations,  is  related  to  the 
molal  reduction-potentials  involved,  b.  Calculate  the  equilibrium- 
constant,  c.  Find  the  composition  of  the  equilibrium  mixture  that 
results  when  metallic  silver  is  placed  in  o.i  f.  Fe(NO3)3  solution.  Ans. 
b,  0.14;  c,  Ag+  or  Fe"1"4",  0.067. 

Prob.  58.  —  Manganates  in  aqueous  solution  undergo  partial  decom- 
position into  permanganates  and  manganese  dioxide  according  to  the 
following  equation:  3MnO4=  +  2H20  =  MnO2(s)  +  2MnO4~+  4OH~. 
Calculate  the  ratio  of  permanganate  to  manganate  in  the  equilibrium 
mixture  at  25°  when  in  that  mixture  the  manganate  is  o.i  formal  and  the 
free  base  is  i  formal.  The  molal  reduction-potentials  at  25°  are  -0.61 
volt  for  the  reaction  MnO4=  +  0=Mn04~,  and  -0.66  volt  for  the 
reaction  Mn02(s)+40H+20  =Mn04=  +2H2O.  Ans.  2.21. 


VOLTAIC  ACTION  AND  POLARIZATION  271 

VI.    VOLTAIC  ACTION,   ELECTROLYSIS,   AND  POLARIZATION 

161.  Concentration-Changes  Attending  Voltaic  Action  and  the 
Resulting  Polarization.  —  Throughout  the  foregoing  considerations, 
as  an  aid  in  evaluating  the  electromotive  force,  it  has  been  assumed 
that  so  large  a  quantity  of  solution  is  present  in  the  cell  that  only 
infinitestimal  concentration-changes  are  produced  in  it  by  the  passage 
of  a  finite  quantity  of  electricity.  The  fact  that  this  is  not  the  case 
in  actual  cells  must  be  taken  into  account,  as  in  the  following  problem. 

Prob.  59.  —  Change  of  Electromotive  Force  due  to  Changes  in  Con- 
centration. —  The  electromotive  force  E  at  15°  of  the  lead  storage-cell 
varies  with  the  mol-fraction  x  of  the  H2SO4  (for  values  of  x  up  to  o.io) 
according  to  the  equation  E  =  1.855  +3-8ox  -  iox2.  Out  of  a  certain  cell 
which  contains  1300  g.  of  10  mol-percent  H2SO4  a  steady  current  of 
5.36  amperes  is  taken  for  10  hours.  Calculate  the  electromotive  force 
of  the  cell  at  the  beginning  and  at  the  end. 

In  practice  the  electromotive  force  is  often  decreased  much  more 
than  these  considerations  indicate  because  the  concentration-changes 
actually  occur  in  the  immediate  neighborhood  of  the  electrodes  and 
are  only  gradually  distributed  by  convection  or  diffusion  through  the 
whole  body  of  the  solution.  Thus  in  the  lead  storage-cell  water  is 
produced  and  acid  is  destroyed  by  the  electrode  reaction  in  the  solu- 
tion impregnating  the  porous  lead-peroxide  electrode,  and  the  acid 
can  only  be  replenished  from  the  main  body  of  the  solution  by  the 
slow  process  of  diffusion.  This  phenomenon  is  one  kind  of  polariza- 
tion, sometimes  called  concentration-polarization;  the  name  polarization 
being  used  in  general  to  denote  the  production  by  the  passage  of  the 
current  of  any  change  in  the  solution  adjoining  the  electrode  or  in 
the  surface  of  the  electrode  which  makes  its  potential  deviate  from 
its  normal  value. 

Concentration-polarization  is  affected  by  various  incidental  condi- 
tions, for  example,  by  agitation  of  the  solution  around  the  electrodes. 
It  is  also  often  greatly  influenced  by  the  current-density,  by  which  is 
meant  the  current  per  unit-area  of  electrode  surface,  thus  the  number 
of  amperes  per  square  centimeter  or  square  decimeter  of  surface  of  a 
specified  electrode. 

Prob.  60.  —  Concentration-Polarization  and  the  Factors  Affecting  It.  — 
When  a  current  is  taken  at  25°  out  of"  a  certain  cell  of  the  form 
Zn(s),  ZnSO4(i  f.),  CuSO4(i  f.),  Cu(s),  the  electromotive  force  soon  be- 
comes fairly  constant  and  remains  so  for  a  time  at  a  value  0.06  volt 
below  its  normal  value,  a.  Show  quantitatively  how  this  might  be 


272  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

accounted  for  by  concentration-polarization,  b.  Explain  how  the 
polarization  would  be  affected  by  increasing  the  current;  by  using 
smaller  electrodes  without  changing  the  current,  thereby  increasing 
the  current-density;  and  by  stirring  the  solutions,  these  being  separated 
from  each  other  by  a  porous  cup. 

162.  Electrolysis  in  Relation  to  Minimum  Decomposition-Potential. 
—  In  order  to  produce  electrolysis  in  any  electrolytic  cell  there  must 
obviously  be  applied  from  an  external  source  an  electromotive  force 
at  least  equal  to  the  electromotive  force  that  is  produced  by  the 
combination  of  solution  and  electrodes,  considering  it  as  a  voltaic 
cell.  The  value  of  the  electromotive  force  that  must  be  applied  to 
compensate  the  electromotive  force  of  the  voltaic  cell  which  develops 
under  the  actual  conditions,  assuming  local  concentration  changes  to 
be  equalized,  may  be  called  the  minimum  decomposition-potential.  Its 
value  can  evidently  be  calculated  in  the  case  of  cells  with  dilute  solu- 
tions by  the  method  considered  in  Art.  155.  It  is  important  to 
recognize  this  fact;  for  the  decomposition-potential  is  sometimes 
treated  as  if  it  were  an  essentially  independent  quantity.  The  actual 
decomposition-potential,  that  is,  the  electromotive  force  that  must  be 
applied  to  produce  continuous  electrolysis,  may  be  much  greater  than 
the  minimum  decomposition-potential  because  of  polarization  effects, 
such  as  are  considered  in  the  next  article. 

Prob.  61.  —  Deposition  of  Metals  by  Electrolysis.  —  A  solution  0.05 
formal  in  H2SO4  and  0.05  formal  in  CuSO4  is  electrolyzed  at  25°  between 
a  mercury  anode  and  a  platinum  cathode.  In  the  mixture  assume  the 
salt  to  be  completely  ionized  and  the  acid  to  be  25%  ionized  into  2H+ 
and  SO4=  and  75%  ionized  into  H+  and  HSO4~.  a.  Calculate  from  the 
molal  electrode-potentials  (taking  that  of  Hg(l)+Hg2SO4(s),  SO4=  to 
be  —0.62  volt)  the  minimum  electromotive  force  which  would  have  to 
be  applied  in  order  to  cause  the  copper  to  deposit,  b.  Find  the  value 
to  which  this  electromotive  force  would  have  to  be  increased  after  99% 
of  the  copper  had  been  precipitated  in  order  that  the  deposition  might 
continue,  assuming  the  ionizations  to  be  the  same  as  in  the  original 
mixture,  c.  Find  the  minimum  electromotive  force  at  which  hydrogen 
could  be  continuously  set  free,  assuming  that  it  attains  at  the  cathode 
an  effective  pressure  of  i  atm. 

Prob.  62.  —  Separation  of  Elements  by  Electrolysis.  —  A  solution 
o.i  formal  in  KC1,  o.i  formal  in  KBr,  and  o.i  formal  in  KI  is  placed, 
together  with  a  platinum  electrode,  in  a  porous  cup;  and  this  is  placed 
within  a  larger  vessel  containing  a  zinc  electrode  and  a  large  quantity 
of  o.i  formal  ZnCl2  solution.  Neglecting  the  liquid-potential  and  any 
polarization  effects,  calculate  the  applied  electromotive  force  required 
at  25°,  a,  to  liberate  99.9%  of  the  iodine;  b,  to  set  bromine  free  at  a 


ELECTROLYSIS  AND  POLARIZATION  273 

concentration  of  o.oooi  molal;  c,to  liberate  99.9%  of  the  bromine  (which 
remains  in  solution);  and  d,  to  liberate  chlorine  at  a  concentration  of 
o.oooi  molal. 

163.  Electrolysis  in  Relation  to  Polarization.  —  In  electrolysis,  as 
in  voltaic  action,  concentration  changes  are  produced  in  the  solutions 
around  the  electrodes;  and  these  changes  have  the  effect  of  increasing 
the  applied  electromotive  force  required.  Thus  in  electrolyzing  a 
copper  solution  between  copper  electrodes  (as  is  done  in  copper  plating) 
only  an  infinitesimal  electromotive  force  is  required  to  start  the 
deposition;  but  the  solution  around  the  cathode  soon  becomes  less 
concentrated  in  copper  and  the  solution  around  the  anode  more  con- 
centrated in  copper,  producing  a  concentration-cell  with  an  electro- 
motive force  opposite  to  that  applied.  This  back  electromotive  force 
would  evidently  be  diminished  by  decreasing  the  current-density,  by 
agitating  the  whole  solution,  or  by  rotating  the  electrodes. 

When  a  gas,  such  as  hydrogen  or  oxygen,  is  set  free  at  an  electrode, 
a  phenomenon,  known  as  gas-polarization,  is  observed  which  does 
not  occur  in  the  deposition  of  metals.  For  when  a  gas  is  involved 
in  the  electrode-reaction  its  partial  pressure  determines  the  electrode- 
potential.  Hence  in  a  cell  exposed  to  the  air  a  slow,  continuous 
electrolysis  may  take  place  before  the  applied  electromotive  force 
equals  the  electromotive  force  of  the  cell  when  the  gas-pressure  is  one 
atmosphere;  for  the  gas-forming  substance  is  produced  on  the  electrode 
at  a  lower  pressure  and  is  dissolved  in  the  solution  and  carried  away 
by  convection  and  diffusion.  Moreover,  a  sudden  increase  in  the  rate 
of  electrolysis  does  not  take  place  when  the  applied  electromotive 
force  is  increased  beyond  that  corresponding  to  a  pressure  of  one 
atmosphere;  for  the  gas-forming  substance  then  passes  into  the  elec- 
trode at  this  higher  pressure,  producing  a  supersaturated  adsorbed 
layer  on  its  surface  or  a  supersaturated  solid  solution  within  the 
metal,  from  which  the  gas  does  not  escape  rapidly  enough  to  reduce  the 
effective  pressure  to  one  atmosphere.  There  is  thereby  produced  a 
back  electromotive  force  which  is  equal  to  the  electromotive  force  of  a 
voltaic  cell  in  which  the  gas  has  this  higher  pressure,  and  which  is 
larger  than  the  electromotive  force  of  a  similar  cell  in  which  the  gas 
has  the  partial  pressure  that  prevails  above  the  solution.  The  amount 
by  which  the  back  electromotive  force  exceeds  the  theoretical  electro- 
motive force  at  the.  prevailing  partial  pressure  is  called  the  polarization 
or  overvoltage. 


274          PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

It  will  thus  be  seen  that  the  applied  electromotive  force  may  be 
considered  resolved  into  three  parts:  one  equal  to  the  minimum 
decomposition-potential  or  to  the  theoretical  electromotive  force  of 
the  resulting  voltaic  cell  at  the  prevailing  partial  pressures;  a  second 
part  equal  to  the  polarization;  and  a  third  part,  that  uncompensated 
by  the  back  electromotive  force  resulting  from  these  two  effects,  and 
therefore  available  for  forcing  a  current  through  the  cell  in  accordance 
with  Ohm's  law.  These  considerations  are  illustrated  by  Prob.  63. 

These  principles  suffice  to  enable  the  phenomena  of  polarization 
to  be  treated  from  experimental  and  thermodynamic  standpoints. 
The  various  hypotheses  that  have  been  proposed  in  regard  to  the 
molecular  mechanism  by  which  gas-polarization  results  can  not  be 
here  considered.  These  hypotheses  involve  considerations  relating 
to  surface-tension,  adsorption,  formation  of  unstable  compounds, 
and  rate  of  reaction  in  solid  phases. 

The  back  electromotive  force  produced  by  the  electrolysis  may  be 
experimentally  determined  in  the  following  ways: 

(1)  By  the  method  described  in  Prob.  63. 

(2)  By  rinding  the  smallest  value  of  the  applied  electromotive 
force  at  which  bubbles  form  slowly  but  continuously  at  the  electrode 
surface  (this  giving  a  value  corresponding  practically  to  a  minimum 
current-density). 

(3)  By  applying  a  definite  electromotive  force  to  the  cell  long  enough 
to  charge  the  electrodes  with  the  decomposition-products  and  then 
short-circuiting  the  electrodes  through  a  high-resistance  potentiometer, 
the  applied  potential  being  at  the  same  time  removed.     From  the 
so-measured  back  electromotive  force  the  overvoltage  is  obtained  by 
subtracting  the  theoretical  electromotive  force  corresponding  to  the 
prevailing  gas-pressure,  which  is  usually  about  one  atmosphere. 

Prob.  63.  —  Gas-Polarization.  —  Electromotive  forces  successively 
increasing  in  magnitude  were  applied  at  22°  to  an  electrolytic  cell  con- 
sisting of  a  large,  unpolarizable,  platinized  platinum  plate  as  anode, 
a  small  mercury  surface  as  cathode,  and  a  o.i  f.  H2SO4  solution 
as  electrolyte.  Hydrogen  gas  at  i  atm.  was  bubbled  steadily  through 
the  cell,  and  a  resistance  of  100,000  ohms  was  placed  in  series  with  it, 
the  resistance  of  the  cell  being  negligible  in  comparison.  The  current- 
strengths  in  millionths  of  an  ampere  corresponding  to  various  applied 
electromotive  forces  in  volts  were  as  follows: 

Current-strength    ....    0.06     0.44     1.20     2.20     3.70     4.82 
Applied  electromotive  force  .    0.32     0.48     0.62     0.77     0.95     1.08 


ELECTROLYSIS  AND  POLARIZATION  275 

a.  Plot  these  current-strengths  as  ordinates  against  the  electromotive 
forces  as  abscissas,  b.  Calculate  the  back  electromotive  force  and 
polarization  corresponding  to  each  of  these  applied  electromotive 
forces,  tabulate  these  values,  and  plot  the  current-strengths  against 
them  on  the  same  diagram,  c.  Calculate  the  effective  pressure  of  the 
hydrogen  at  the  electrode  corresponding  to  the  smallest  of  these  back 
electromotive  forces,  assuming  the  logarithmic  relation  valid  for  perfect 
gases  to  hold  up  to  this  pressure. 

The  overvoltage  is  always  found  to  increase  with  increase  of  the 
applied  electromotive  force  and  of  the  current-density;  but  it  varies 
in  a  highly  specific  way  with  the  chemical  nature  of  the  gas  and  with 
the  chemical  nature  and  physical  state  of  the  metallic  electrode. 
Thus  in  certain  experiments  made  with  i  normal  H2SO4  the  over- 
voltage  of  the  hydrogen  was  found  to  have  the  following  values: 

Small  Large 

current-density       current-density 

Platinized  platinum      .      .     .  o.oo  volt  0.07  volt 

Smooth  platinum    ....  0.03  0.65 

Lead 0.36  1.23 

Mercury 0.44  1.30 

In  certain  experiments  with  2  normal  KOH  solution  with  a  moderate 
current-density  the  overvoltage  of  the  oxygen  was  found  at  the  start 
to  be  0.44  volt  on  platinized  platinum,  0.84  volt  on  smooth  platinum, 
and  0.50  volt  on  iron;  the  value  increasing  to  1.46  on  smooth  platinum 
and  to  0.59  on  iron  after  two  hours'  passage  of  the  current. 

The  phenomenon  of  gas-polarization  and  the  overvoltage  attending 
it  are  of  great  significance  in  technical  processes.  The  overvoltage 
may  greatly  diminish  the  energy-efficiency  of  the  process,  the  energy- 
efficiency  being  the  ratio  of  the  minimum  electrical  energy  theoretically 
required  to  produce  a  definite  quantity  of  some  product  of  the  elec- 
trolysis to  the  energy  actually  expended.  Overvoltage  may  also 
make  processes  practicable  which  would  otherwise  not  be  possible; 
thus  in  charging  a  lead  storage-cell  hydrogen  is  not  set  free  at  the  lead 
owing  to  its  overvoltage,  although  the  potential  of  the  half-cell 
H2(i  atm.),  H2SO4.2oH2O  is  about  0.4  volt  less  than  that  of  the  half- 
cell  Pb+PbSO4(s),  H2SO4.2oH20. 

Prob.  64.  —  Energy-Efficiency  and  Overvoltage.  —  In  a  certain  com- 
mercial alkali-chlorine  cell  sodium  hydroxide  and  chlorine  are  produced 
at  an  iron  cathode  and  graphite  anode,  respectively,  by  the  continuous 
electrolysis  of  a  25%  NaCl  solution  which  is  slowly  flowed  through  a 
diaphragm  from  the  anode  to  the  cathode  compartment  (to  prevent  the 
hydroxide-ion  from  migrating  to  the  anode).  In  a  certain  case  4.5  volts 


276  PRODUCTION  OF  WORK  IN  VOLTAIC  CELLS 

were  applied  to  the  cell,  whose  resistance  was  0.00080  ohm,  yielding  a 
current  of  2000  amperes;  and  there  flowed  off  each  hour  from  the 
cathode  27,460  g.  of  solution  containing  10%  of  NaOH  and  13%  of  NaCl. 
The  electromotive  force  of  the  voltaic  cell  H2(i  atm.),  NaOH(io%) 
+  NaCl(i3%),  NaCl(25%),  Cl2(i  atm.)  has  been  independently  deter- 
mined to  be  2.3  volts,  a.  Calculate  the  current-efficiency  in  the 
production  of  the  sodium  hydroxide,  b.  Calculate  the  energy-efficiency 
of  its  production,  c.  Calculate  the  voltage  used  to  produce  the  current 
and  that  used  to  overcome  the  polarization,  and  tabulate  these  together 
with  the  minimum  decomposition-potential.  A ns.  b,  47.0%. 


CHAPTER  XII 

THE  EFFECT  OF  TEMPERATURE  ON  THE   WORK   PRO- 
DUCIBLE  BY  ISOTHERMAL   CHEMICAL   CHANGES 
AND   ON  THEIR  EQUILIBRIUM   CONDITIONS 


I.    THE  FUNDAMENTAL   SECOND-LAW  EQUATION 

164.  The  Quantity  of  Work  Producible  from  a  Quantity  of  Heat 
that  Passes  from  One  Temperature  to  Another.  —  This  chapter  is 
mainly  devoted  to  a  consideration  of  the  effect  of  temperature  on  the 
free-energy  changes  that  attend  isothermal  changes  in  state.  To  this 
effect  is  closely  related,  in  virtue  of  the  free-energy  relations  already 
derived,  the  effects  of  temperature  on  the  equilibrium  of  chemical 
changes  and  on  the  electromotive  force  of  voltaic  cells.  Before  these 
effects  can  be  properly  considered  the  more  general  thermodynamic 
relation  referred  to  in  the  title  of  this  article  must  be  known.  This 
will  now  be  derived. 

The  second  law  of  thermodynamics  has  already  been  expressed 
(in  Art.  135)  in  fundamental  form  by  the  statement  that  a  process 
whose  final  result  is  only  a  transformation  of  heat  into  work  is  an 
impossibility;  and  it  was  shown  that,  in  accordance  with  this  state- 
ment, when  work  is  produced  in  surroundings  of  constant  tempera- 
ture, there  is  always  a  change  in  the  state  of  the  system  employed 
for  the  transformation.  Chapters  X  and  XI  have  been  devoted  to 
a  determination  of  the  quantities  of  work  producible  by  various 
isothermal  changes  in  state.  There  will  now  be  considered  the  other 
conditions,  involving  difference  in  temperature  in  the  surroundings, 
under  which  the  second  law  permits  that  work  be  produced  out  of 
heat  by  systems  which  undergo  no  permanent  change  in  state. 

When  a  quantity  of  heat  is  transformed  into  work  by  a  cyclical 
process  (that  is,  by  a  process  in  which  the  system  undergoes  a  cycle 
of  changes  in  state),  an  additional  quantity  of  heat  is  always  taken 
up  from  surroundings  at  a  higher  temperature  and  given  out  to  sur- 
roundings at  a  lower  temperature.  That  is  to  say,  even  when  a  differ- 
ence of  temperature  exists,  only  a  fraction  of  the  heat  taken  up  by  the 
system  from  the  warmer  surroundings  can  be  transformed  into  work. 
Important  questions  at  once  arise  as  to  what  determines  the  fraction 
that  can  be  so  transformed  —  whether  it  is  dependent  on  the  nature  of 
the  process  employed,  and  how  it  varies  with  the  temperatures. 

277 


278        EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM 

In  order  to  determine  whether  the  quantity  of  work  that  can  be 
produced  when  any  definite  quantity  of  heat  is  transferred  by  a 
cyclical  process  from  a  higher  to  a  lower  temperature  is  dependent 
upon  the  nature  of  the  system  employed  for  the  transformation,  or 
upon  the  way  in  which  the  transformation  is  carried  out,  assume 
that  two  different  reversible  cyclical  processes,  carried  out  with 
different  systems  or  in  a  different  way  with  the  same  system,  could 
produce  two  unequal  quantities  of  work  by  transferring  an  equal 
quantity  of  heat  from  a  higher  to  a  lower  temperature.  Then  cause 
the  process  that  produces  the  larger  quantity  of  work  W"  to  take 
place  in  such  a  way  that  it  takes  up  a  quantity  of  heat  Qi  at  the 
higher  temperature  7\,  transfers  a  part  of  it  Qz  to  the  lower  tempera- 
ture T2,  and  transforms  the  remainder  into  work  W";  and  cause  the 
other  process,  which  in  transferring  the  same  quantity  of  heat  Q%  from 
TI  to  Tz  produces  the  smaller  amount  of  work  W,  to  take  place  in  the 
reverse  direction  —  that  is,  so  that  it  takes  up  the  heat  Qz  transferred 
by  the  former  process  to  the  lower  temperature,  and  raises  it  to  the 
higher  temperature  by  expending  the  required  amount  of  work  W. 
It  is  then  evident  that  the  net  result  of  these  operations  would  be  the 
production  of  a  quantity  of  work  W"—W  from  an  equivalent 
quantity  of  heat  without  any  other  change  having  been  brought  about 
either  in  the  systems  or  in  the  surroundings.  Since  this  is  contrary 
to  the  fundamental  statement  of  the  second  law,  the  supposition  made 
that  the  two  processes  produce  unequal  quantities  of  work  is  unten- 
able. This  important  conclusion  may  be  explicitly  stated  as  follows : 
the  quantity  of  work  which  can  be  produced  when  a  definite  quantity 
of  heat  is  transferred  from  one  temperature  to  another  by  any  process 
in  which  the  system  employed  undergoes  no  permanent  change  in 
state  is  not  dependent  on  the  nature  of  the  process. 

By  the  conclusion  just  reached  the  determination  of  the  relation 
between  the  temperatures  and  the  proportion  of  heat  transformable 
into  work  is  greatly  facilitated;  for  evidently  it  is  now  only  necessary 
to  determine  what  that  relation  is  for  a  single  reversible  cyclical 
process.  Such  a  process  is  considered  in  the  following  problem. 

Prob.  i.  —  Derivation  of  the  Second-Law  Equation.  —  Consider  the 
following  cyclical  process  carried  out  reversibly  with  N  mols  of  a 
perfect  gas  contained  in  a  cylinder  closed  with  a  weighted,  frictionless 
piston.  Start  with  the  gas  at  the  temperature  T+dT  and  volume  Vi. 
(i)  Keeping  the  gas  in  a  large  heat-reservoir  at  T+dT,  diminish  gradu- 
ally the  weight  on  the  piston  and  cause  the  gas  to  expand  till  its  volume 


THE  SECOND-LAW  EQUATION  279 

becomes  v2.  (2)  Fix  the  piston  so  that  the  volume  must  remain  con- 
stant, and  place  the  gas  in  a  large  heat-reservoir  at  a  temperature  T. 
(3)  Keeping  the  gas  in  the  heat-reservoir  at  the  temperature  T,  com- 
press it  by  releasing  the  piston  and  gradually  increasing  the  weight 
upon  it  until  the  volume  %  of  the  gas  has  been  restored  to  its  original 
value  VL  (4)  Fix  the  piston  again  so  as  to  keep  the  volume  constant, 
and  place  the  gas  in  a  heat-reservoir  at  the  temperature  T+dT. 
a.  Represent  this  process  by  a  lettered  diagram  in  which  the  ordinates 
denote  temperatures  and  the  abscissas  volumes,  b.  Formulate  an 
expression  for  the  quantities  of  work  Wi,  Wz,  Wz,  W^  and  for  the 
quantities  of  heat  Qi,  Qz,  Q3,  Q*,  produced  in  the  surroundings  in  the 
separate  steps  of  this  process,  taking  into  account  the  principles  con- 
sidered in  Arts.  24,  25,  and  26.  c.  Formulate  a  relation  between  the 
quantity  of  work  2TF  produced  in  the  whole  process  and  the  quantity 
of  heat  Q  imparted  to  the  reservoir  at  the  temperature  T. 

Since  it  was  previously  shown  that  the  second  law  requires  that  the 
same  quantity  of  work  be  produced  when  a  definite  quantity  of  heat 
is  transferred  by  any  reversible  cyclical  process  whatever  from  one 
definite  temperature  to  another,  it  is  evident  that  the  equation  de- 
rived in  the  preceding  problem  for  one  such  process  is  an  exact  expres- 
sion of  the  second  law  for  every  such  process.  This  equation  there- 
fore expresses  one  of  the  fundamental  principles  of  physical  science. 

The  equation  just  derived,  which  will  be  called  simply  the  second- 
law  equation,  may  be  written  in  the  form: 


In  this  equation  S  W  denotes  the  algebraic  sum  of  all  the  quantities 
of  work  produced  in  any  reversible  cyclical  process  taking  place  at 
two  temperatures  T  and  T  +  dT  in  which  the  quantity  of  heat  Q  is 
transferred  to  the  surroundings  at  the  temperature  T.  The  work- 
quantity  2W  is  obviously  infinitesimal  in  correspondence  with  the 
infinitesimal  temperature-difference  d  T. 

Prob.  2.  —  General  Principles  Deduced  from  the  Second-Law  Equa- 
tion. —  Show  that  the  second-law  equation  leads  to  the  following 
conclusions:  a.  When  there  is  no  difference  of  temperature  in  the 
surroundings  heat  cannot  be  transformed  at  all  into  work  by  any 
cyclical  process.  b.  In  order  to  carry  heat  from  a  lower  to  a  higher 
temperature  work  must  be  withdrawn  from  the  surroundings,  c.  The 
fraction  of  the  heat  transformable  into  work  for  a  given  difference  in 
temperature  is  greater  the  lower  the  temperature,  d.  When  the  lower 
temperature  is  the  absolute  zero  heat  can  be  completely  transformed 
into  work. 


280        EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM 

II.    THE   EFFECT  OF  TEMPERATURE   ON  THE   EQUILBRIUM  OF 
UNIVARIANT   SYSTEMS 

165.  Effect  of  Temperature  on  the  Pressure  at  which  the  Phases 
of  Univariant  Systems  are  in  Equilibrium.  —  •  An  important  relation 
derivable  from  the  second  law  of  thermodynamics  is  that  named  in  the 
title  of  this  article.  It  is  expressed  by  the  Clapeyron  equation: 


= 

dT  ~  T  Av  ' 

In  this  equation  dp  denotes  the  increase  produced  by  a  temperature- 
increase  dT  in  the  pressure  at  which  the  phases  of  a  univariant  system 
are  in  equilibrium  at  the  temperature  T,  and  AH  and  Av  denote  the 
increases  in  heat-content  and  in  volume  which  attend  the  conver- 
sion at  the  temperature  T  and  at  the  equilibrium-pressure  p  of  any 
definite  quantity  of  substance  from  the  state  in  which  it  exists  in 
one  phase  or  set  of  phases  into  the  state  in  which  it  exists  in  the 
other  phase  or  set  of  phases.  For  example,  the  temperature-co- 
efficient dp/dT  of  the  dissociation-pressure  p  of  solid  calcium  car- 
bonate at  any  temperature  T  can  be  calculated  by  substituting  in 
the  Clapeyron  equation  for  AH  the  increases  in  heat-content  and 
for  Av  the  increase  in  volume  that  attend  the  change  in  state 
CaCO3(s)  =  CaO(s)  -f-  CO2(at  p)  at  the  temperature  T. 
This  equation  will  now  be  derived  from  the  second-law  equation. 

Prob.  j.  —  Derivation  of  the  Clapeyron  Equation.  —  Consider  the 
following  cyclical  process:  Start  with  some  definite  quantity  of  a 
substance  (for  example,  iH2O)  existing  as  a  phase  (for  example,  liquid 
water)  which  is  in  equilibrium  with  a  second  phase  (for  example,  ice) 
at  the  temperature  T  and  pressure  p.  (i)  Cause  the  substance  to  pass 
under  these  equilibrium  conditions  from  the  first  phase  (liquid  water) 
in  which  its  volume  is  vi  into  the  second  phase  (ice)  in  which  its  volume 
is  %;  (2)  heat  the  substance  to  T+dT,  whereby  the  pressure  becomes 
p+dp  and  the  volume  v2+dv2;  (3)  cause  it  to  pass  from  the  second 
phase  (ice)  into  the  first  phase  (liquid  water)  at  T-\-dT  under  the 
equilibrium  pressure  p+dp,  whereby  the  volume  becomes  v\-\-dv\;  (4) 
cool  the  substance  to  T,  whereby  it  reverts  to  its  original  state. 
a.  Represent  this  process  by  a  diagram  in  which  the  ordinates  denote 
pressures  and  the  abscissas  volumes,  b.  Formulate  the  quantities  of 
work,  Wi,  W2,  W3,  Wt,  produced  in  the  four  steps  of  the  process  when 
it  is  carried  out  reversibly,  summate  these  quantities  of  work,  cancel 
the  differentials  of  the  second  order,  substitute  the  result  in  the  second- 
law  equation,  and  replace  the  heat  quantity  Q  in  the  equation  by  its 
equivalent  —AH.  c.  Specify  the  change  in  state  to  which  the  heat- 
content  decrease  —  AH  refers. 


UNIVARIANT  SYSTEMS  281 

In  applications  of  the  Clapeyron  equation  a  definite  change  in 
state  should  first  be  formulated,  and  then  A#  and  Az>  should  be  evalu- 
ated in  accordance  with  it,  expressing  the  heat-quantity  A#  and  the 
work-quantity  dp  AD  in  the  same  units.  When  one  of  the  phases  in- 
volved is  a  gas  at  small  pressure,  Av  may  be  determined  by  neglecting 
the  volume  of  the  solid  or  liquid  phases  and  expressing  the  volume  of 
the  gaseous  phase  in  terms  of  its  temperature  and  pressure  with  the 
aid  of  the  perfect-gas  law.  Applications  of  the  equation  to  systems  of 
this  kind  (those  involving  liquid  and  gaseous  phases)  were  considered 
in  Art.  33.  Other  applications  are  illustrated  by  the  following  problems. 
Effect  of  Pressure  on  Melting  Point.  — 

Prob.  4.  —  Derive  from  the  Clapeyron  equation  a  principle  expressing 
the  direction  of  the  effect  of  pressure  on  the  melting-point  of  solid  sub- 
stances, taking  into  account  the  fact  that  fusion  is  always  attended  by 
an  absorption  of  heat. 

Prob.  5.  —  Calculate  the  variation  per  atmosphere  of  the  melting- 
point  of  ice.  At  o°  and  i  atm.  its  density  is  0.91 7  and  the  heat  of  fusion 
of  i  g.  is  79.7  cal. 

Prob.  6.  —  Effect  of  Pressure  on  Transition-Temperature.  —  a.  Calcu- 
late the  variation  per  atmosphere  of  the  transition- temperature  (95.5°) 
of  rhombic  into  monoclinic  sulfur  from  the  densities  of  the  two  forms, 
which  are  2.07  and  1.96  respectively,  and  from  the  fact  that  the  transi- 
tion of  i  at.  wt.  of  sulfur  from  the  rhombic  into  the  monoclinic  form  is 
attended  at  95.5°  by  an  absorption  of  105  cal.  b.  State  what  this 
shows  as  to  the  slope  of  the  line  BE  in  the  sulfur  diagram  of  Art.  115; 
and  what  conclusion  as  to  the  slope  of  the  line  CF  can  be  drawn  by 
considering  the  density  of  liquid  sulfur,  which  is  1.81  at  115°. 

Prob.  7.  —  Heat  of  Hydration  derived  from  the  Vapor-Pressure  of  Salt- 
Hydrates.  — With  the  aid  of  the  temperature- vapor-pressure  curves  of 
the  diagram  of  Art.  120  derive  approximate  values  of  the  heat-effect 
attending  the  reaction  Na2HPO4.7H2O(s)-f-5H2O(g)  =Na2HPO4.i  2H20(s) 
at  25°. 

In  order  to  integrate  the  Clapeyron  equation  exactly,  A#  and  Afl 
must  be  expressed  as  functions  either  of  the  equilibrium-temperature 
or  of  the  equilibrium-pressure.  When  the  system  consists  of  only 
solid  and  liquid  phases  and  when  only  moderate  changes  of  pressure 
(such  as  20  atm.)  are  involved,  AH,  Av,  and  T  in  the  second  member  of 
the  equation  may  be  considered  constant  in  approximate  calculations. 
When  one  of  the  phases  consists  of  a  perfect  gas,  &v  may  be  expressed 
as  a  temperature-function  by  means  of  the  perfect-gas  equation.  The 
heat-quantity  AH  may  always  be  so  expressed  in  terms  of  the  heat- 
capacities  of  the  substances  involved,  in  the  way  described  in  Art.  131. 


282        EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM 

Applications  of  Integrated  Forms  of  the  Clapeyron  Equation.  — 

Prob.  8.  —  Calculate  the  melting-point  of  ice  at  20  atm.  with  the  aid 
of  the  data  of  Prob.  5. 

Prob.  9.  —  State  what  heat-quantity  can  be  calculated  from  the  dis- 
sociation-pressures of  solid  NH4SH  into  NH3  and  H2S,  which  are  500 
mm.  at  25°  and  182  mm.  at  10°;  and  calculate  its  value,  assuming  it  to 
be  constant  through  the  temperature  interval  involved. 

*Prob.  10.  —  Calculate  the  vapor-pressure  of  solid  iodine  at  25°  from 
the  following  data:  Its  vapor-pressure  at  100°  is  47.5  mm.,  its  molal 
heat  of  vaporization  at  100°  is  —14,600  cal.,  its  atomic  heat-capacity  is 
6.7  between  25°  and  100°,  and  the  molal  heat-capacity  of  its  vapor  at 
constant  pressure  is  7.8  between  those  temperatures. 

*The  derivation  of  the  Clapeyron  equation  shows  that  it  determines 
the  equilibrium  conditions  of  any  type  of  system  in  which  an  iso- 
thermal change  in  state  can  take  place  under  a  constant  equilibrium- 
pressure,  which  is  determined  only  by  the  temperature;  for  evidently 
in  all  such  cases,  and  only  in  such  cases,  the  expressions  for  the  work- 
quantities  involved  in  the  cyclical  process  by  which  the  equation  was 
derived  will  have  the  same  form.  An  isothermal  change  can,  how- 
ever, take  place  without  any  change  in  the  pressure  only  when  the 
pressure  is  determined  solely  by  the  temperature  and  is  not  changed 
by  any  transfer  of  substance  from  one  phase  of  the  system  to  another. 
In  other  words,  the  Clapeyron  equation  is  applicable  to  all  systems, 
and  only  to  systems,  which  are  actually  or  in  effect  univariant.  Thus 
it  has  been  shown  applicable  to  one-component  systems  existing  in 
two  phases,  like  ice  and  water  or  water  and  water-vapor,  or  to  two- 
component  systems  existing  in  three  phases,  like  CaCOs,  CaO,  and 
CO2.  It  is  also  applicable,  however,  to  systems  that  have  been  made 
in  effect  univariant  by  the  specification  that  the  composition  of  each 
of  the  phases  present  shall  remain  constant  when  the  isothermal 
change  takes  place  and  when  the  temperature  of  the  system  is  varied. 
Thus  the  equation  can  be  applied  to  a  two-component  system  existing 
in  only  two  phases,  like  a  solution  and  its  vapor,  provided  such  a 
change  in  state  be  considered  that  the  solution  remain  constant  in 
composition  when  some  of  the  solvent  vaporizes  out  of  it,  as  in  the 
case  when  an  infinite  quantity  of  the  solution  is  subjected  to  the 
process.  Hence  the  equation  can  be  used  to  determine  the  change  in 
the  vapor-pressure  of  a  solution  with  the  temperature.  Similarly,  it 
can  be  applied  to  a  two-component  system  consisting  of  a  solution  and 
a  solid  phase,  like  a  melted  binary  alloy  in  contact  with  one  of  the  solid 


UNIVARIANT  SYSTEMS 


283 


metals,  so  as  to  determine  the  effect  of  pressure  on  the  temperature 
at  which  the  solid  metal  will  separate  from  a  melt  of  the  specified 
composition. 

*Prob.  ii.  —  Application  of  the  Clapeyron  Equation  to  Solutions.  — 

a.  In  applying  the  Clapeyron  equation  to  determine  the  effect  of  temper- 
ature on  the  vapor-pressure  of  a  10%  NaCl  solution,  specify  the  change 
in  state  to  which  the   quantities  A#   and    A0   would    correspond. 

b.  State  what  equilibrium  could  be  studied  by  applying  the  Clapeyron 
equation  to  a  system  consisting  of  solid  sodium  chloride  and  its  satu- 
rated solution,  and  specify  the  change  in  state  to  which  the  quantities 
Aff  and  Az>  would  correspond. 


284       EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM 

III.     THE  EFFECT  OF  TEMPERATURE  ON  THE  EQUILIBRIUM  OF  CHEMICAL 
REACTIONS   IN   GENERAL 

166.  The  Effect  of  Temperature  on  the  Free-Energy  Decrease 
Attending  Any  Isothermal  Change  in  State.  —  By  applying  the  funda- 
mental second-law  equation  to  certain  reversible  cyclical  processes 
there  can  be  derived,  as  shown  in  Probs.  12  and  13,  the  following 
differential  equations  expressing  the  change  with  the  temperature  of 
the  decreases  —AA  and  — AF,  denned  as  in  Art.  136,  in  the  work- 
content  and  in  the  free  energy  of  a  system  attending  any  isothermal 
change  in  its  state. 

d(-AA)       AU-AA  ,  , 

W 


dT  T 

d(-AF)      Aff-AF 


(2) 


dT  T 

These  equations  are  general  expressions  of  the  second  law  in  forms 
especially  convenient  in  considerations  relating  to  chemical  equi- 
librium and  electromotive  force.  It  is  important  fully  to  appreciate 
the  significance  of  the  quantities  occurring  in  them  and  the  case  to 
which  each  equation  is  applicable.  These  are  shown  by  the  derivations 
of  the  equations  (in  Probs.  12  and  13)  to  be  as  follows.  The  quantities 
— A  A  and  —  AF  denote  the  decreases  in  the  work-content  and  in  the 
free  energy  of  the  system  which  attend  any  isothermal  change  in  its 
state  at  the  temperature  T',  and  AU  and  AH  denote  the  accompanying 
increases  in  its  energy-content  and  heat-content.  The  differential 
quantities  d(—  AA)  and  d(—AF)  signify  that  when  the  same  change 
in  state  takes  place  at  T+dT,  instead  of  at  T,  it  is  attended  by  a 
work-content  decrease  of  —  AA+d(—  AA)  and  by  a  free-energy  de- 
crease of  -AF+d(-AF),  instead  of  one  of  -AA  and  of  -AF.  The 
work-content  equation  is  applicable  to  cases  where  the  initial  volume, 
and  also  the  final  volume,  of  the  system  is  the  same  at  the  two  tem- 
peratures; and  the  free-energy  equation  is  applicable  to  cases  where 
the  initial  pressure,  and  also  the  final  pressure,  is  the  same  at  the  two 
temperatures. 

In  accordance  with  the  practice  adopted  throughout  the  preceding 
considerations,  only  the  free-energy  equation  will  hereafter  be  em- 
ployed. For  purposes  of  integration  this  equation  is  more  conve- 
niently written  in  the  following  forms: 

-AFX    Ari. 


CHEMICAL  REACTIONS  IN  GENERAL  285 

'T          =     I 

T\       JTV 

This  equation  in  any  of  its  forms  will  be  called  the  second-law  free- 
energy  equation. 

Derivation  of  the  Work-Content  and  Free-Energy  Equations.  — 
Prob.  12.  —  To  derive  equation  (i)  given  in  the  above  text,  consider  a 
reversible  cyclical  process  involving  the  following  steps:  (i)  Any 
change  in  state  of  any  system  at  the  temperature  T,  by  which  a  quantity 
of  work  W  is  produced;  (2)  a  change  in  the  temperature  of  the  system 
at  constant  volume  from  T  to  T+dT;  (3)  a  change  in  the  state  of  the 
system  at  the  temperature  T+dT  which  is  the  reverse  of  the  change 
in  state  in  the  first  step,  this  change  being  attended  by  a  production  of  a 
quantity  of  work  —  (W + dW)',  (4)  a  change  in  the  temperature  of  the 
system  at  constant  volume  from  T+dT  to  T.  a.  Represent  this  pro- 
cess by  a  lettered  diagram  in  which  the  ordinates  denote  temperatures 
and  the  abscissas  volumes,  b.  Find  an  expression  for  ZJF  for  this 
process,  c.  By  substituting  it  in  the  second-law  equation  and  by 
making  other  appropriate  substitutions,  show  that  equation  (i) 
results. 

Prob.  13.  —  To  derive  equation  (2)  given  in  the  above  text,  consider  a 
reversible  cyclical  process  just  like  that  described  in  the  preceding 
problem,  except  that  in  steps  (2)  and  (4)  the  system  is  kept  at  constant 
pressure,  instead  of  at  constant  volume,  a.  Represent  this  process 
by  a  lettered  diagram  in  which  the  ordinates  denote  temperatures  and 
the  abscissas  volumes,  designating  the  pressures  and  volumes  of  the 
system  at  the  beginning  of  each  of  the  four  steps  by  (i)  pi  and  v\\  (2) 
pz  and  v2',  (3)  p2  and  v2+dv2',  and  (4)  pi  and  v\+dv\.  b.  Find  an  expres- 
sion for  2W  for  this  process,  c.  By  substituting  this  expression  in  the 
second-law  equation  and  making  other  appropriate  substitutions  based 
on  the  definitions  of-AF  and  -AH  given  in  Arts.  136  and  128,  show 
that  equation  (2)  results,  d.  Show  that  the  free  energy  equations  (2) 
and  (3)  are  identical  by  carrying  out  in  the  first  member  of  equation 
(3)  the  indicated  differentiation  and  by  making  simple  transforma- 
tions. 

167.  The  Effect  of  Temperature  on  the  Equilibrium  of  Chemical 
Changes  Involving  Perfect  Gases.  —  By  substituting  in  the  second-law 
free-energy  equation  (3)  of  Art.  166  the  expression  derived  in  Arts.  140 
and  142  for  the  free-energy  decrease  attending  an  isothermal  chemical 
change  between  perfect  gases  or  between  solid  substances  and  perfect 
gases,  and  noting  that  the  initial  and  final  pressures  occurring  in 
this  expression  must  not  vary  with  the  temperature  if  this  equation 
(3)  is  to  be  applicable,  there  is  obtained  the  following  equation,  com- 
monly called  the  van't  Hof  equation,  expressing  the  effect  of  tempera- 


286        EFFECT  OF   TEMPERATURE  ON  EQUILIBRIUM 

ture  on  the  equilibrium  pressures  pA,  pz,  etc.,  and  on  the  equilibrium- 
constant  K  of  the  chemical  reaction: 


Prob.  14.  —  Derivation  of  the  Van't  Hoff  Equation.  —  Derive  in  the 
way  indicated  in  the  text  the  van't  Hoff  equation  from  the  general 
expression  given  in  Art.  140  for  the  free-energy  decrease  attending  any 
chemical  change  between  gases. 

Prob.  15.  —  Qualitative  Principle  Expressing  the  Effect  of  Temperature 
on  Chemical  Equilibrium.  —  Derive  from  the  van't  Hoff  equation  a 
principle  showing  how  the  direction  in  which  an  equilibrium  is  dis- 
placed by  increase  in  temperature  is  related  to  the  sign  of  the  heat- 
effect  attending  the  reaction. 

Applications  of  the  Qualitative  Principle.  — 

Prob.  16.  —  Show  how  the  equilibrium  hi  a  gaseous  mixture  of  Ck, 
HC1,  O2,  and  H2O  at  25°  would  be  displaced  by  increasing  the  tempera- 
ture. The  heat  of  formation  at  25°  of  iHCl(g)  is  22,000  cal.  and  that 
of  iH2O(g)  is  57,800  cal. 

Prob.  17.  —  The  dissociation-pressure  of  solid  CaCO3  (and  of  all  other 
solid  substances  which  dissociate  into  one  or  more  gaseous  products) 
increases  with  rising  temperature.  State  whether  heat  is  absorbed  or 
evolved  when  the  dissociation  takes  place  at  constant  temperature. 

In  order  to  integrate  the  van't  Hoff  equation,  the  heat-content 
increase  MI  must  be  expressed  as  a  function  of  the  temperature. 

This  temperature  function  is  usually  derived,  in  the  way  shown  in 
Art.  131,  from  a  knowledge  of  the  heat-content  increase  at  some  one 
temperature  and  of  the  heat-capacities  at  constant  pressure  of  the 
substances  involved  in  the  reaction.  Thus  the  function  may  always 
be  obtained  by  integrating  the  following  differential  equation,  in 
which  AC  represents  the  difference  between  the  heat-capacity  (equal 
to  CE+CF.  .)  of  the  system  in  its  final  state  and  its  heat-capacity 
(equal  to  CA-f-CB.  .)  in  its  initial  state: 


The  integration  can  evidently  be  carried  out  when  AC  is  known  to  be 
constant  or  when  it  can  be  expressed  as  a  function  of  the  tempera- 
ture, as  illustrated  by  the  following  problems. 

Integration  of  the  Van't  Hoff  Equation.  — 

Prob.  18.  —  a.  Derive  the  expression  d(AZ7)=AC.dr  in  the  way 
described  hi  Art.  131.  b.  Integrate  it  for  the  case  that  AC  is  a  function 
of  the  form  AC=ACo+ar+j3r2,  where  AC0,  a,  and  0  are  constants. 

Prob.  ig.  —  Find  a  numerical  expression  for  AZ7  for  the  reaction 


CHEMICAL  REACTIONS  IN  GENERAL  287 

2CO(g)  +O2(g)  =  2CO2(g)  from  the  heat-capacity  data  given  in  Art.  27 
and  in  Prob.  29  of  Art.  134  and  the  heats  of  formation  of  iCO(g)  and 
iCO2(g)  at  20°,  which  are  25,900  and  94,200,  respectively. 

Prob.  20.  —  Integrate  the  van't  Hoff  equation  between  the  limits 
Ti  and  T2,  KI  and  K2,  for  the  following  cases:  a,  when  AC  is  zero  and 
therefore  A#  does  not  vary  with  the  temperature;  b,  when  AC  has  a 
finite  value  which  does  not  vary  with  the  temperature;  c,  when  AC 
varies  with  the  temperature  in  the  way  considered  in  Prob.  i8b. 

In  numerical  applications  of  the  van't  Hoff  equation,  in  order  to 
guard  against  errors  in  the  sign  of  the  heat-content  increase  and  in 
its  value  with  respect  to  the  multiple  chosen,  the  reaction  under  con- 
sideration should  first  be  formulated  in  a  definite  chemical  equation 
and  then  values  of  K,  AH,  and  AC  should  be  adopted  in  conformity 
with  it.  For  the  heat-capacities  of  some  important  gaseous  sub- 
stances and  of  elementary  solid  substances  see  Arts.  27  and  55. 

Applications  of  the  Van't  Hoff  Equation  to  the  Equilibrium  of  Gas 
Reactions.  — 

Prob.  21.  —  It  has  been  found  that  when  dry  air  (containing  21.0 
mol-percent  of  oxygen  and  78.0  mol-percent  of  nitrogen)  is  kept  at 
1957°  till  equilibrium  is  reached,  4.3%  of  the  oxygen  present  is  con- 
verted into  nitric  oxide.  Calculate  the  percentage  that  would  be  so 
converted  at  3000°.  The  formation  of  iNO(g)  from  its  elements  at 
20°  is  attended  by  a  heat-absorption  of  21,600  cal.  For  the  heat- 
capacities  of  the  three  gases  see  Art.  27.  Ans.  18.5%. 

Prob.  22.  —  Formulate  an  exact  numerical  expression  by  which  the 
dissociation  7  of  carbon  dioxide  into  carbon  monoxide  and  oxygen  at 
any  temperature  T  and  any  total  pressure  p  can  be  calculated.  Its 
dissociation  at  1205°  and  i  atm.  is  0.032%.  Refer  to  Prob.  19  for  the 
heat  data  needed. 

Prob.  23.  —  When  a  mixture  of  0.49  mol  of  O2  and  i.oo  mol  of  HC1  is 
kept  at  386°  and  i  atm.  in  contact  with  solid  cuprous  chloride  (which 
acts  as  a  catalyst)  till  equilibrium  is  reached,  80%  of  the  HC1  is  con- 
verted into  C12  and  H2O.  a.  Formulate  the  reaction,  and  calculate  its 
equilibrium-constant  at  386°.  b.  Formulate  as  a  function  of  the 
absolute  temperature  the  increase  in  heat-content  attending  this  re- 
action, using  the  heat-content  data  of  Prob.  16  and  the  heat-capacity 
data  of  Art.  27  and  of  Prob.  17,  Art.  131.  c.  Derive  from  these  results 
a  numerical  expression  for  the  equilibrium-constant  at  25°. 

Prob.  24.  —  Calculate  from  the  heat  data  given  below  and  the 
values  of  the  heat-capacities  given  in  Art.  27:  0,  the  heat  of  dis- 
sociation of  iHgO(s)  into  mercury  vapor  and  oxygen  at  357°;  6,  a 
temperature-function  expressing  the  heat-content  increase  attending 
this  dissociation  above  357°;  c,  the  dissociation-pressure  of  solid  mercuric 
oxide  at  357°.  The  heat  of  formation  of  iHgO(s)  at  20°  is  21,700  cal. 


288        EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM 

The  vaporization  of  iHg(l)  at  its  boiling-point  (357°)  absorbs  14,160 
cal.  The  mean  value  of  the  atomic  heat-capacity  of  liquid  mercury 
between  20°  and  357°  is  6.36  cal.  per  degree;  and  the  heat-capacity  of 
iHgO(s)  may  be  taken  as  10.87  at  all  temperatures.  The  dissociation- 
pressure  of  mercuric  oxide  at  390°  is  180  mm.  Ans.  a,  35,500  cal.; 
c,  70  mm. 

168.  The  Effect  of  Temperature  on  the  Equilibrium  of  Chemical 
Changes  Involving  Perfect  Solutes.  —  By  substituting  in  the  second-law 
free-energy  equation  of  Art.  166  the  expression  given  in  Art.  141  for  the 
free-energy  decrease  attending  any  chemical  change  between  perfect 
solutes,  and  noting  that  in  that  expression  the  last  term  containing 
the  initial  and  final  molalities  does  not  vary  with  the  temperature 
when  the  solution  is  heated  at  constant  pressure,  there  again  results 
the  van't  Hoff  equation,  now  expressed  in  terms  of  the  equilibrium 
molalities: 


This  equation  is  applicable  also  to  chemical  changes  between  solid 
substances  and  solutes  at  small  concentrations,  since  it  was  shown  in 
Art.  142  that  the  participation  of  solid  substances  in  the  change  does 
not  affect  the  expression  for  the  free  energy.  It  applies  also  to  changes 
in  which  the  solvent  (thus  the  water  in  aqueous  solutions)  enters  into 
reaction  with  solutes  at  small  concentrations. 

From  this  equation,  as  from  the  corresponding  one  in  terms  of 
pressures,  can  be  derived  the  important  qualitative  principle  that  the 
equlibrium  of  a  chemical  change  is  displaced  by  increase  of  tempera- 
ture in  that  direction  in  which  the  reaction  is  attended  by  an  increase 
in  heat-content  (or  by  an  absorption  of  heat). 

The  integration  of  the  equation  evidently  involves  the  expression 
of  the  heat-content  as  a  function  of  the  temperature.  Since  in  the 
case  of  solutions  the  temperature  interval  often  is  not  large,  the  in- 
crease in  heat-content  can  frequently  be  regarded  as  constant;  and  it 
is  to  be  so  regarded  in  the  following  problems  unless  otherwise  stated. 
When  this  is  not  admissible  its  variation  with  the  temperature  must 
be  known.  This  must  usually  be  derived  from  direct  determinations 
of  the  heat-effects  attending  the  reaction  at  two  or  more  different 
temperatures;  for  there  is  likely  to  be  a  large  error  involved  in  cal- 
culating it  from  the  partial  heat-capacities  of  the  solutes,  since  they 
form  only  a  small  part  of  the  total  heat-capacity  of  the  solution. 


CHEMICAL  REACTIONS  IN  GENERAL  289 

Prob.  25.  —  Application  of  the  Qualitative  Principle.  —  State  the 
conclusions  in  regard  to  the  heat  of  solution  that  can  be  drawn  from  the 
facts  that  solubility  of  all  gaseous  substances  is  decreased,  and  the  solu- 
bility of  most  solid  substances  is  increased,  by  an  increase  of  tempera- 
ture. 

Prob.  26.  —  Effect  of  Temperature  on  the  lonization  of  Water.  —  The 
neutralization  of  largely  ionized  univalent  acids  and  bases  in  fairly 
dilute  solution  has  been  found  by  direct  measurements  at  2°,  10°,  18°, 
26°,  and  34°  to  evolve  14,750  —  52^  cal.  at  the  centigrade  temperature  /. 
The  value  of  the  ionization-constant  of  water  at  25°  is  i  .00X10-". 
Calculate  its  value  at  o°.  Ans.  o.i  Xio-14. 

Prob.  27.  —  Effect  of  Temperature  on  the  Hydrolysis  of  Salts.  —  Calcu- 
late the  hydrolysis  of  o.i  f.  NH4CN  at  o°  from  its  hydrolysis  (47%)  at 
25°  and  from  the  fact  that  on  mixing  at  25°  a  solution  containing  o.2NH| 
and  loco  g.  of  water  with  one  containing  0.2HCN  and  1000  g.  of 
water  and  bringing  the  mixture  back  to  25°  there  is  a  heat  evolution 
of  152  cal. 

Prob.  28.  —  Effect  of  Temperature  on  Solubility.  —  The  solubility  of 
silver  chloride  in  water  is  i.io  X  io-5  formal  at  20°  and  15.2  X  io~6  formal 
at  100°.  a.  State  just  what  heat-quantity  can  be  computed  from  these 
data,  and  calculate  its  value,  b.  State  by  what  thermochemical 
measurement  it  could  be  determined  experimentally. 


290       EFFECT  OF  TEMPERATURE  ON  EQUILIBRIUM 

IV.  THE  EFFECT  OF  TEMPERATURE  ON  THE  ELECTROMOTIVE 
FORCE  OF  VOLTAIC  CELLS 

169.  The  Effect  of  Temperature  on  the  Electromotive  Force  of 
Voltaic  Cells.  —  By  substituting  in  the  second-law  free-energy  equa- 
tions of  Art  166  the  quantity  ENF  shown  in  Art  145  to  represent  the 
free-energy  decrease  attending  any  change  in  state  taking  place  reversi- 
bly  hi  a  voltaic  cell  there  result  the  following  expressions,  known  as 
the  Gibbs-Helmholtz  equation,  for  the  effect  of  temperature  on  the 
electromotive  force  of  voltaic  cells: 

fa 


In  this  equation  A£T  denotes  the  increase  in  heat-content  which 
attends  the  change  in  state  that  takes  place  when  N  faradays  of  elec- 
tricity flow  through  a  cell  of  electromotive  force  E  containing  infinite 
quantities  of  the  constituent  substances. 

Effect  of  Temperature  on  Electromotive  Force.  — 

Prob.  29.  —  Show  from  the  Gibbs-Helmholtz  equation  under  what 
conditions  the  electromotive  force  of  a  cell,  a,  is  independent  of  the  tem- 
perature; b,  is  proportional  to  the  absolute  temperature;  c,  increases 
with  rising  temperature;  d,  decreases  with  rising  temperature. 

Prob.  jo.  —  a.  Calculate  the  temperature-coefficient  at  25°  of  the 
electromotive  force  of  the  cell  H2(i  atm.),  H2SO4(o.oi  f.),  O2(i  atm.), 
which  was  found  in  Prob.  20,  Art.  149,  to  be  1.228  volts,  b.  Calculate 
the  electromotive  force  of  this  cell  at  o°,  assuming  that  the  heat-effect  at- 
tending the  change  in  state  does  not  vary  appreciably  between  o  and  25°. 

Prob.  31.  —  Calculate  the  temperature-coefficient  at  25°  of  the  electro- 
motive force  (1.262  volts)  of  the  cell  H2(i  atm.),  HC1(4  f.),  Cl2(i  atm.) 
from  the  heat-content  (-22,000  cal.)  of  iHCl(g)  and  its  heat  of  solu- 
tion (16,000  cal.)  in  4  f.  HC1  solution. 

Prob.  32.  —  Calculate  the  electromotive  force  at  40°  of  the  storage-cell 
Pb(s)+PbS04(s),  H2S04.ioH2O,  PbS04(s)+PbO2(s),  from  its  electro- 
motive force  (2.096  volts)  at  o°  and  the  heat  data  needed.  The  heat- 
contents  at  18°  of  iPbSO4(s),  iPbO2(s),  and  iH2SO4(l)  are  -216,210 
cal.,  —62,900  cal.,  and  —192,900  cal.,  respectively.  The  heats  of 
solution  at  18°  of  iH2SO4(l)  and  of  iH2O(l)  in  H2SO4.ioH2O  are  12,820 
cal.  and  230  cal.,  respectively. 

Prob.  33.  —  Calculation  of  Heat  of  Reaction.  —  The  electromotive 
force  of  the  cell  Hg(l)  +Hg2Cl2(s),  KCl(o.oi  f.)  +KNO3  (i  f  ),KNO3(i  f.) 
+KOH(o.oi  f.),  Hg20(s)+Hg(l)  at  o°  is  0.1483  volt  and  at  18.5°  is 
0.1636  volt.  a.  Formulate  the  change  in  state  that  takes  place  in  the 
cell  when  two  faradays  pass  through  it.  b.  Calculate  the  attendant 
increase  in  its  heat-content. 


ELECTROMOTIVE  FORCE  OF  VOLTAIC  CELLS    291 

*It  will  have  been  noted  that  the  calculation  of  the  increase  in  heat- 
content  attending  chemical  changes  in  voltaic  cells  involves  the  heat- 
effect  produced  by  the  addition  of  definite  quantities  of  the  solute  or 
solvent  to  an  infinite  quantity  of  the  solution  at  a  specified  concen- 
tration. This  kind  of  heat-effect  often  has  to  be  considered  also  in 
evaluating  the  increase  in  heat-content  used  in  calculating  the  change 
with  the  temperature  of  the  equilibrium  conditions  of  chemical  sys- 
tems, as  in  the  application  of  the  Clapeyron  equation  to  solutions  in 
Art.  165.  It  is  called  (because  of  the  relation  derived  in  Prob.  356) 
the  partial  heat  of  solution  of  the  substance.  It  can  be  derived  from 
calorimetrically  measured  heats  of  dilution,  like  those  considered  in 
Art.  132,  in  the  ways  illustrated  by  the  following  problems. 

* Determination  of  Partial  Heats  of  Solution.  — 

*Prob.  34.  —  The  measured  heat-effect  Q  attending  the  mixing  of 
iH2SO4(l)  with  iVH2O4(l)  at  18°  is  expressed  by  the  empirical  equation, 
Q  =  i7&6oN/(N  +  i.8o)  for  values  of  N  not  exceeding  20.  Calculate 
the  heat-effect  attending  the  addition  to  an  infinite  quantity  of 
H2SO4.ioH2O,  a,  of  iH2O(l);  b,  of  iH2SO4(l).  Am.  a,  231;  b,  12820. 

*Prob.  35.  —  a.  Determine  with  the  aid  of  a  plot  of  the  heats  of  dilu- 
tion given  in  the  table  of  Art.  132  the  heat -effect  attending  the  addition 
of  iH20(l)  to  an  infinite  quantity  of  ZnCl2.  iooH2O.  b.  Derive  a  relation 
betweenthe  heat-effect  at  tending  the  dissolving  of  iZnCl2(s)iniooH20(l), 
that  attending  the  addition  of  iooH2O(l)  to  an  infinite  quantity  of 
a  solution  of  the  composition  iZnC!2.iooH2O,  and  that  attending  the 
addition  of  iZnC!2(s)  to  an  infinite  quantity  of  such  a  solution. 
c.  Calculate  the  value  of  the  last  named  heat-effect,  d.  Calculate  the 
heat-content  of  iZnC!2  in  iZnC!2.iooH20,  taking  that  of  iZnC!2(s) 
equal  to— 97,30003!. 

*Prob.  36.  —  The  E/ect  of  Temperature  on  Electrode-Potentials.  — 
a.  Specify  precisely  the  change  in  state  whose  heat-effect  would  be 
used  in  calculating  the  temperature-coefficient  of  the  molal  electrode- 
potential  of  Zn(s),  Zn4^.  b.  Calculate  a  value  of  this  temperature- 
coefficient  from  the  heat  values  found  in  Prob.  35,  the  heat-content 
(-22,000  cal.)  of  iHCl(g),  and  its  heat  of  dilution  as  given  in  Art. 
132,  considering  for  this  purpose  solutions  with  iooH2O  per  iZnC!2, 
or  per  iHCl  as  infinitely  dilute. 


CHAPTER  XIII 
*SYSTEMIZATION  OF  FREE-ENERGY  VALUES 


170.  Importance  of  Systemizing  Free-Energy  Values.  —  Since  a 
knowledge  of  the  changes  in  free  energy  attending  chemical  changes 
makes  it  possible  to  predict  the  direction  in  which  they  can  take  place 
and  the  conditions  under  which  they  are  in  equilibrium,  the  working- 
out  of  a  complete  system  of  free-energy  values  becomes  an  experi- 
mental problem  of  great  chemical  importance.    The  more  general 
principles  by  which  changes  in  free  energy  can  be  determined  have 
already  been  fully  considered  in  the  preceding  chapters.    The  pur- 
poses of  this  chapter  are  the  direct  application  of  these  principles  to 
the  determination  of  free-energy  values,   and  the  presentation  of 
certain  conventions  by  which  free-energy  values  can  be   system- 
atized.   These  conventions  will  be  fir-st  considered. 

171.  Expression  of  Free-Energy  Changes  by  Equations.  —-  Since  the 
free  energy  of  a  substance  is  a  quantity  which,  like  its  heat-content, 
is  fully  determined  by  its  state,  changes  in  free  energy  can  be  repre- 
sented by  equations  corresponding  in  every  respect  to  those  used  for 
expressing  changes  in  heat-content;  conventions  being  adopted  for 
showing  the  state  of  the  substance  identical  with  those  described  in 
Art.  129.    It  will  suffice  to  supplement  the  specifications  of  that  article 
by  the  following  statements. 

Free-energy  values  are  commonly  expressed  in  calories  on  account 
of  their  close  relations  with  heat-content  data,  which  are  commonly 
so  expressed. 

The  pressure  at  which  each  substance  exists  must  always  be  definite, 
since  the  free-energy  is  in  general  in  a  high  degree  dependent  on  the 
pressure.  When  not  specified,  the  pressure  is  understood  to  be  one 
atmosphere. 

In  correspondence  with  the  convention  as  to  the  arbitrary  zero- 
point  of  the  heat-content  scale,  the  free  energy  at  any  definite  tem- 
perature of  any  definite  quantity  of  a  substance  in  any  definite  state  is 
equal  to  the  increase  in  free  energy  that  attends  the  formation  of  the 
substance  in  that  state  out  of  the  pure  elementary  substances  at  the 
same  temperature,  and  at  a  pressure  of  one  atmosphere,  each  elemen- 
tary substance  being  in  the  state  of  aggregation  that  is  the  stable  one 

292 


SYSTEMIZATION  OF  FREE-ENERGY   VALUES          293 

at  this  temperature  and  pressure.  Thus  the  free  energy  of  iHI(i  atm.) 
at  25°  refers  to  its  formation  from  gaseous  hydrogen  and  solid 
iodine  at  25°  and  one  atmosphere;  but  at  400°  it  refers  to  its  forma- 
tion from  gaseous  hydrogen  and  gaseous  iodine  at  400°  and  one 
atmosphere. 

The  free  energy  of  a  substance  in  a  solution  is  understood  to  mean  the 
increase  in  free  energy  attending  the  formation  of  the  pure  substance 
out  of  the  elementary  substances  plus  that  attending  the  introduction 
of  it  into  an  infinite  quantity  of  the  solution  under  consideration.  This 
free  energy  is  indicated  by  attaching  to  the  chemical  formula  a  paren- 
thesis showing  the  concentration  or  composition  of  the  solution;  for 
example,  iKCl(at  o.i  f.),  iH2SO4  (in  H2SO4.ioH2O).  The  free  energy 
of  water  in  a  solution  is  often  conveniently  referred  to  that  of  pure 
water  (instead  of  to  that  of  hydrogen  and  oxygen)  as  zero;  and  when 
so  referred  the  symbol  Aq  (instead  of  H2O)  is  used;  for  example,  5Aq 
(in  H2SO4.ioH2O). 

The  free  energy  of  an  ion  in  aqueous  solution  is  conveniently  de- 
fined, in  correspondence  with  the  definition  of  electrode-potentials 
given  in  Art.  151,  as  the  increase  in  free  energy  that  attends  the  forma- 
tion of  the  ion  in  solution  at  the  specified  concentration  out  of  the 
elementary  substance  at  one  atmosphere,  hydrogen-ion  at  i  molal 
being  formed  at  the  same  time  out  of  hydrogen  gas  at  one  atmosphere. 
Thus  the  free  energy  of  Pb++  (o.i  m.)  is  equal  to  the  increase  in  free 
energy  that  attends  the  reaction  Pb(s)  +  2H+(i  m.)=Pb++(o.i  m.) 
+H2(i  atm.) ;  the  free  energy  of  H+(i  m.)  as  well  as  that  of  H2(i  atm.) 
being  thereby  arbitrarily  assumed  to  be  zero. 

When  necessary,  equations  expressing  changes  in  free  energy  may 
be  distinguished  from  those  expressing  changes  in  heat-content  by 
prefixing  to  them  the  symbols  (F)  and  (H),  respectively;  and  the 
absolute  temperature  may  be  shown  by  attaching  to  this  letter  a 
subscript .  For  example : 

(F298)        H2(g)  +  JOsfe)  =  H20(l)  +  56,620  cal. 

(#298)        H2(g)  +  i02(g)  =  H20(l)  +  68,400  cal. 

(F298)    H20(l)  =  - 56,620  cal.  (#298)    H20(l)  =  - 68,400 cal. 

Since  the  free  energy  of  a  substance  at  various  small  pressures  or 
concentrations  can  be  calculated,  by  the  logarithmic  expressions  of 
Art.  137  and  138,  from  the  free  energy  at  any  one  definite  small  pressure 
or  concentration,  it  suffices  to  determine  and  record  the  free  energy 


2Q4         SYSTEMIZATION  OF  FREE-ENERGY   VALUES 

of  each  gaseous  substance  at  one  atmosphere  and  of  each  solute  in 
each  solvent  at  i  molal.  These  standard  values  are,  however,  com- 
monly so  expressed  as  to  represent  the  free  energy  of  the  substance 
at  this  pressure  or  concentration  under  the  assumption  that  it  behaves 
as  a  perfect  gas  or  perfect  solute  up  to  this  pressure  or  concentration; 
that  is,  they  are  calculated  by  the  logarithmic  equation  from  those 
determined  at  much  smaller  pressures  or  concentrations. 

In  the  case  of  solutes  so  concentrated  that  they  show  large  deviations 
from  the  behavior  of  perfect  solutes  the  free  energies  must  be  deter- 
mined at  a  series  of  concentrations,  and  these  values  must  be  separately 
recorded.  Or,  if  preferred,  the  activity-coefficients  corresponding 
to  the  free  energies  at  various  formalities  of  the  substance  may  be 
calculated  and  recorded. 

172.  Determination  of  the  Free  Energies  of  Substances.  —  With 
the  aid  of  the  principles  relating  to  free  energy  considered  in  chapters 
X,  XI,  and  XII,  and  the  conventions  presented  in  the  preceding 
article,  the  free  energies  of  substances  in  their  different  states  can  be 
determined,  as  illustrated  by  the  following  problems. 

Prob.  i.  —  Free  Energies  of  Elementary  Substances  in  Different  States. 
With  the  aid  of  the  principles  of  Arts.  137-139,  calculate  the  free 
energy  at  25°  of,  a,  I2(g.  at  i  atm.);  b,  I2(at  i  m.  in  H2O);  and  c,  I2  (at 
i  m.  in  CC14).  At  25°  the  vapor-pressure  of  pure  iodine  is  0.305  mm., 
its  solubility  in  water  is  0.00132  molal,  and  its  distribution-ratio  between 
carbon  tetrachloride  and  water  is  86. 

Prob.  2.  —  Free  Energy  of  Solutes  at  High  Concentrations.  — 
a.  Calculate  the  free  energy  of  iNH3  in  a  8.6  n.  solution  in  water  at  25° 
from  the  following  data  at  25°.  The  free  energy  of  iNH3(g)  is—  4740 
cal.  The  vapor-pressure  of  NH3  from  a  0.2  n.  solution  in  water  is 
2.70  mm.  The  distribution-ratio  of  NH3  between  CC14  and  H2O  is 
0.0040  when  the  NH3  is  0.2  n.  in  the  water,  and  0.0086  when  the  NH3 
is  8.6  n.  in  the  water,  b.  State  the  principles  involved,  c.  State  how 
the  same  free  energy  might  be  obtained  from  another  kind  of  measure- 
ment with  the  8.6  n.  aqueous  solution. 

Free  Energies  Derived  from  Electromotive  Forces.  — 

Prob.  j.  —  Calculate  the  free  energy  of  iHCl(g)  at  25°  from  the  facts 
that  at  25°  the  cell  H2(i  atm.),  HC1(4  f.),  Cl2(i  atm.)  has  an  electro- 
motive force  of  1.262  volts,  and  that  the  vapor-pressure  of  the  HC1  in  its 
4  formal  solution  is  0.0182  mm.  Ans.  —22,800  cal. 

Prob.  4.  —  Calculate  the  free  energy  of  iAgCl(s)  at  25°  from  the  molal 
electrode-potentials  and  the  solubility  of  AgCl  which  is  i.3oXio~6 
formal  at  25°.  Ans.  —  26,200  cal. 

Prob.  5.  —  Calculate  the  free  energy  of  iH20(l)   at  25°  from  the 


SYSTEMIZATION  OF  FREE-ENERGY   VALUES         295 

electromotive  force  (0.926  volts)  of  the  cell  H2(g),  NaOH(o.i  f.), 
HgO(s)  H-Hg(l)  and  from  the  free  energy  (-13880  cat.)  of  iHgO(s)  at 
25°.  Ans.  —  56,600  cal. 

Prob.  6.  —  Calculate  the  free  energy  of  iAg2O(s)  at  25°.  The  electro- 
motive force  of  the  cell  H2(g),  KOH(o.i  f.),  Ag2O(s)  +Ag(s>  is  1.172 
volt  at  25°.  Ans.  —2500  cal. 

Prob.  7.  —  Formulate  the  free  energy  equation  which  can  be  derived 
from  the  molal  electrode-potential  of,  a,  Cl2(g),  Cl~,  and  b,  O2(g),  OH~. 
Find  the  free  energy  at  25°,  c,  of  Cl~  (i  m.),  and  d,  of  OH~(i  m.).' 
Ans.  c,  —31,400  cal.;  d,  —37,400  cal. 

Free  Energies  Derived  from  Equilibrium  Measurements.  — 
Prob.  8.  —  Calculate  the  free  energy  of  iNO(g)  at  1957°  from  the  data 
of  Prob.  21,  Art,  167.    Ans.  13,700  cal. 

Prob.  9.  —  Calculate  the  free  energy  of  iHgO(s)  at  357°  from  its  dis- 
sociation pressure  at  357°  found  in  Prob.  24,  Art.  167.  Ans.  —  5700  cal. 

'The  Free  Energy  of  a  Substance  at  One  Temperature  Derived  from  Us 
Free  Energy  at  Another  Temperature.  — 

Prob.  10.  —  With  the  aid  of  the  free-energy  equation  of  Art.  166  cal- 
culate the  free  energy  of  iNO(g)  at  25°  from  its  free  energy  at  1957° 
found  in  Prob.  8  and  from  the  heat  data  of  Prob.  21  of  Art.  167.  Ans. 
2  0,500  cal. 

Prob.  ii.  —  Calculate  the  free  energy  of  iHgO(s)  at  25°  from  its  free 
energy  at  357°  found  in  Prob.  9  and  from  the  heat  data  of  Prob.  24, 
Art.  167,  considering  the  heat  of  dissociation  of  solid  mercuric  oxide 
into  liquid  mercury  and  oxygen  to  be  a  linear  function  of  the  tempera- 
ture between  25°  and  357°.  Ans.  —14,100  cal. 

Prob.  12.  —  Calculate  the  free  energy  at  25°  of  i  at.  wt.  of  monoclinic 
sulfur  from  the  facts  that  its  conversion  into  rhombic  sulfur  at  its 
transition-point  (95.5°)  is  attended  by  a  heat-evolution  of  105  cal.  and 
that  the  atomic  heat-capacities  of  the  monoclinic  and  rhombic  forms 
are  6.0  and  5.7  cal.  per  degree.  Tabulate  this  result  with  that  obtained 
in  Prob.  19  of  Art.  139. 

Prob.  13.  —  Derivation  of  Free  Energies  from  Original  Data.  —  By  a 
thorough  examination  of  the  chemical  literature  to  find  all  the  data 
bearing  on  the  problem  and  by  a  critical  consideration  of  these  data 
derive  the  best  values  of  the  free  energy  at  25°,  and  of  a  function  ex- 
pressing its  change  with  the  temperature,  for  one  of  the  following  solid 
substances  (assigned  by  the  instructor):  Ag20,  HgO,  AgCl,  PbCl2, 
Hg2Cl2,  HgCl2,  CuCl,  Agl,  etc. 

173.  The  Equilibrium-Conditions  of  Chemical  Reactions  Derived 
from  the  Free  Energies  of  the  Substances  Involved.  —  When  values 
of  the  free  energies  of  substances  have  been  once  determined  by  the 
methods  already  described,  these  values  can  be  employed,  conversely, 
for  calculating  the  free-energy  decrease  attending  chemical  changes 


296        SYSTEMIZATION  OF  FREE-ENERGY   VALUES 

and  their  equilibrium-constants,  as  in  the  following  problems.  From 
this  fact  arises  the  great  importance,  already  referred  to,  of  a  complete, 
systematic  knowledge  of  the  free  energies  of  substances. 

The  Equilibrium-Constants  of  Chemical  Reactions  Calculated  from  the 
Free  Energies  of  the  Separate  Substances.  — 

Prob.  14.  —  a.  Calculate  from  the  free  energy  values  already  con- 
sidered the  equilibrium-constant  at  25°  of  the  reaction: 

Ag2O(s)  +H2O+2C1-  =  2AgCl(s)  +2OH- 

b.   State  what  mixture  would  finally  result  if  iAg2O(s)  were  treated  at 
25°  with  1 1.  of  o.i  f.  Nad  solution. 

Prob.  15.  —  a.  Calculate  from  the  free  energy  values  the  equilibrium- 
constant  of  the  reaction  4HCl(g)+O2(g)  =2Cl2(g)+2H2O(g)  at  25°. 
b.  Calculate  the  mol-percents  of  chlorine  and  oxygen  in  the  gas  which 
at  i  atm.  and  25°  would  escape  from  a  4  f .  HC1  solution  if  oxygen  at 
i  atm.  were  passed  through  it  in  contact  with  a  catalyst  so  that  equi- 
librium was  established.  At  25°  in  4  f.  HC1  solution  the  vapor-pres- 
sures of  the  water  and  HC1  are  19.6  mm.  and  0.0182  mm.,  respectively. 
Am.  a,  1.3X10";  6,  C12,  7.2%,  O2,  90%. 


NOTATION 


a  * 

^ 

activity;  area. 

n 

number. 

a,  b 

van  der  Waals  constants. 

n 

number  of  molecules. 

c 

molal  concentration;  mola- 

n 

number  of  molecules  in  one  mol 

lity. 

P 

pressure. 

d 

density. 

r 

radius. 

d 

differential. 

s 

solubility. 

e 

base  of  natural  logarithms. 

t 

time. 

f 
S 

force, 
gravity-acceleration. 

t 
u 

centigrade  temperature, 
velocity. 

h 

height;  hydrolysis. 

V 

volume. 

i 

mol-number. 

V 

specific  volume. 

k 

kinetic-energy  constant. 

V 

molal  volume. 

k 

specific  reaction-rate. 

X 

mol-fraction. 

I 

distance;  length. 

x,y,z 

coordinates. 

m 

mass. 

A 

atomic  weight  ;  work-content        M 

molecular  weight. 

A,B 

constants. 

N 

number  of  mols. 

C 

heat-capacity. 

P 

osmotic  pressure. 

C 

specific  heat-capacity. 

P 

probability. 

C 

molal  heat-capacity. 

Q 

heat  evolved. 

E 

energy. 

R 

gas-constant. 

F 

free  energy. 

S 

entropy. 

H 

heat-content. 

T 

absolute  temperature. 

K 

equilibrium-constant. 

U 

energy-content. 

K 

kinetic  energy. 

W 

work. 

c 

normal  concentration;  nor- 

L 

specific  conductance. 

mality. 

N 

number   of   equivalents   or 

E 

electromotive  force. 

faraday  s. 

E 

electrode-potential. 

Q 

quantity  of  electricity. 

E 

molal  electrode-potential. 

R 

resistance. 

F 

faraday. 

T 

transference  number. 

I 

current-strength. 

U 

ion-mobility. 

L 

conductance. 

V 

potential. 

a 

activity-coefficient, 
coefficients. 

X 
A 

wave-length, 
equivalent  conductance. 

7 

dissociation  or  ionization. 

V 

frequency. 

A 

increment. 

7T 

circumference-diameter  ratio. 

V 

viscosity. 

S 

summation. 

297 

INDEX 


In  this  index  formal  statements  of  laws  or  principles  and  their  formula- 
tions in  equations  are  itemized  under  the  heading  "laws,  principles,  funda- 
mental equations."  Definitions  of  "concepts,  properties,  and  fundamental 
constants,"  values  of  the  "constants  of  chemical  substances,"  and  defini- 
tions and  values  of  "units"  are  entered  under  these  headings,  respectively. 


ACIDS  and  Bases, 

dibasic  acids,  ionization,  157-159. 

displacement  from  salts,  159-160. 

heat  of  ionization,  220. 

heat  of  neutralization,  219-220. 

ionizatioh,  118,  123,  152. 

titration,  electrometric,  267. 

titration  with  indicators,  162-164. 
Acid  salts,  ionization,  158-159. 
Activity  and  activity-coefficient, 

applications,  172-173. 

change  with  concentration,  171-172. 

concept,  169,  170. 

determination,  170-171,  234,  250. 

relation  to 

electromotive  force,  250. 
free  energy,  233-234,  237. 
solubility,  173. 
vapor-pressure,  170. 
Adiabatic  expansion,  41,  44. 
Adsorption  in  contact  catalysis,  142. 
Alloys,          / 

annealihg  or  tempering,  197. 

compound  formation,  193-194, 199. 

cooling-curves,  194,  198,  202. 

diagrams,  194,  196,  199,  201. 

eutectic  point  and  composition,  192, 

193,   202. 

freezing-point  lowering,  193. 

solid  solutions,  198-200. 

ternary,  202. 
Annealing,  197. 
Arc  process  of  nitrogen  fixation,  136, 

287. 

Association,  molecular,  63,  71. 
Atomic  and  molecular  theories, 

gaseous  state,  12,  22-34,  41-46. 

general  discussion,  5,  95. 

liquid  state,  51-54,  64. 

(See  also  Ionic  theory.) 
Atomic  weights, 

determination,  6,  17,  18,  42,  94,  95. 

relation  to  combining  weights,  6,  7, 
17,  18. 

table  of  values,  8. 
Avogadro's  principle,  12,  24. 


BASES.    See  Acids  and  bases. 
Bessemer  steel  process,  15. 
Boiling-point 

composition  diagrams,  68,  73. 

concentrated  solutions,  67-69,  72- 

75- 

constant,  determination,  62. 
raising,  relation  to 

composition,  60-63. 
vapor-pressure,  59-60. 
relation  to 

barometric  pressure,  48. 
heat  of  vaporization,  49,  60, 

61. 

vapor-pressure,  48,  59-61. 
Boyle's  law, 

derivation,  23. 
statement,  10. 

CARRIER  catalysis,  140-141. 
Catalysis  and  catalysts, 

carriers,  140-141. 

contact  agents,  141. 

enzymes,  143. 

hydrogen-ion    and    hydroxide-ion, 
142. 

mechanism,  140-142. 

solvents,  178-179. 

water,  143. 

Cells,  voltaic.    See  Voltaic  cells. 
Chemical 

formulas,  8,  18. 

principles,  3. 

substances, 

classes,  3,  4,  6. 
relation  to  Henry's  law,  64. 
relation  to  Raoult's  law,  58. 
Clapeyron  equation, 

applications,  49,  60,  81,  281-283. 

approximate,  50. 

derivation,  kinetic,  54. 

derivation,  thermodynamic,  280. 

formulation,  49,  50,  280. 

integration,  50,  281-282. 
Clark  standard  cell,  262. 
Cohesion  of  gases,  27,  28,  46- 

299 


3oo 


INDEX 


Color-change 

of  indicators,  161. 

on  neutralization,  160. 
Combining  volume  of  gases,  12. 
Combining  weights, 

determination,  6,  7. 

law,  4,  5. 

Combustion,  heat  of,  209. 
Complex-salt  formation,  65,  78,  in,  167, 

169,  203,  267. 
Components, 

concept,  174,  181-183. 

determination,  183. 

(See  Phase  equilibria.) 
Composition  by  weight,  4-9,  55,  56. 
Compound  formation 

in  solid  alloys,  193-194, 199. 

in  solutions,  71,  72. 
Compression  of  liquids,  231. 
Concentrated  solutions, 

boiling-point,  67-68,  73~7S- 

freezing-point,  82-83. 

osmotic  pressure,  87-88. 

vapor-pressure,  67-72. 
Concentration 

by  hot  gases,  222. 

cells,  250-251. 

polarization,  271,  273. 

CONCEPTS,  PROPERTIES,  FUNDAMENTAL 

CONSTANTS, 

absolute  temperature,  u. 
acceleration  of  gravity,  13. 
activity,  169. 
activity-coefficient,  170. 
additive  property,  93. 
adiabatic,  41. 
adsorption,  142. 
ampere,  100. 
anion,  100. 
anode,  98. 
atmosphere,  13. 
atom,  5. 

atomic  energy,  43. 
atomic  weight,  6,  8,  17. 
Avogadro  number,  25. 
bimolecular  reactions,  133. 
bivariant  systems,  180. 
boiling-point,  48. 
boiling-point  constant,  61. 
boiling-point,  molal  raising,  62. 
carrier  catalysts,  140. 
catalysis,  140. 
cathode,  98. 
cation,  100. 
chemical  substance,  6. 
chemistry,  3. 
cohesion-constant  a,  27. 


CONCEPTS,  PROPERTIES,  FUNDAMENTAL 

CONSTANTS  (continued) 
cohesion  pressure,  27. 
combining  weight,  4. 
component,  174,  181-182. 
compound  substance,  4. 
compression-coefficient,  231. 
concentrated  solutions,  56,  67. 
concentration,  55,  56. 
concentration-cell  ,250. 
conductance,  actual,  112. 
conductance,  equivalent,  112. 
conductance-ratio,  117,  119. 
conductance,  specific,  112. 
cooling-curves,  192. 
current-density,  271. 
cycle  of  changes,  223. 
cyclical  process,  277. 
decomposition-potential,  272. 
density,  10. 
dissociation,  16. 
dissociation-constant,  148. 
dissociation-pressure,  165. 
distribution-ratio,  77. 
electrode,  98. 
electrode-potential,  256. 
electrode-potential,  molal,  257. 
electrolyte,  98. 
electrolysis,  98. v 
electron,  101. 
electronic  charge,  101. 
element,  4. 

elementary  substance,  4. 
energy,  35. 
energy-efficiency,  275 
enzyme,  143. 
equation  of  state,  21. 
equilibrium,  chemical,  145. 
equilibrium-constant,  146,  148. 
equivalent  weight,  9. 
eutectic  mixture,  192. 
eutectic  point,  192. 
eutectic  ternary,  202. 
faraday,  100. 
formality,  55. 
formula- weight,  8. 
fractionation,  3. 
free  energy,  227,  292. 
free  path,  mean,  30. 
freezing-point,  80. 
freezing-point-constant,  81. 
freezing-point,  molal  raising,  82. 
gas-constant,  13. 
gases,  perfect,  10. 
gas-polarization,  273. 
heat,  36. 

heat-capacity,  molal,  39. 
heat-capacity-ratio,  41. 
teat-capacity,  specific,  39. 


INDEX 


301 


CONCEPTS,  PROPERTIES,  FUNDAMENTAL 

CONSTANTS  (continued) 
.    heat-content,  210,  213. 
heat  of 

dilution,  218. 

fusion,  217. 

reaction,  209. 

solution,  217. 
partial,  291. 

transition,  217.    ^ 

vaporization,  217. 
humidity,  190. 
hydrate,  65. 
hydrolysis,  154. 
hydrolysis-constant,  156. 
indicator,  161. 
indicator-constant,  161. 
ion,  96,  100. 
ion-conductance,  120. 
ion-constituents,  100. 
ionic  theory,  96. 
ionization,  96. 
ionization-constant,  152. 
irreversible  process,  224. 
isomorphism,  91. 
Joule-Thomson  effect,  44. 
kinetic  energy,  24. 
kinetic-energy  constant  k,  25. 
kinetic  hypotheses,  22. 
liquid-potential,  254. 
mixture,  3. 
mobility  of  ions,  107. 
mobility  of  ion-constituents,  107. 
molal 

concentration,  56* 

properties,  10. 

quantities,  21. 

weight,  12. 
molality,  55. 
molecular 

composition,  18. 

formula,  18. 

weight,  12. 
molecule,  5. 
mol-fraction,  15. 
mol-number,  96. 
mol-ratio,  55. 
normal  concentration,  56. 
normality,  55. 
osmotic  pressure,  84. 
overvoltage,  273. 
partial  pressure,  14. 
perpetual  motion 

first  kind,  35. 

second  kind,  76,  223. 
phase,  55. 
phase  rule,  181. 
physical  chemistry,  3. 


CONCEPTS,  PROPERTIES,  FUNDAMENTAL 

CONSTANTS  (continued) 
physics,  3. 
polarization,  273. 
porous-plug  experiment,  44. 
pressure,  13,  37. 
probability-function,  33. 
pure  substance,  3. 
reaction 

order,  133. 

rate,  131. 

specific  rate,  131. 

uni-,  bi-,  tri-  molecular,   132, 

133- 

reduction-potential,  270. 
resistance,  112. 
reversible  process,  224. 
salting-out  effect,  66. 
semipermeable  wall,  14. 
solid  solutions,  197. 
solubility,  64. 
solubility-constant,  64. 
solubility-product,  167. 
solute,  55. 
solution,  55. 
solvate,  65. 
solvent,  55. 
specific  heat,  39. 
specific  quantities,  21. 
state  of  system,  35,  174. 
substance,  chemical,  6. 
substance,  pure,  3. 
system,  35. 

temperature,  absolute,  n. 
temperature,  centigrade,  n. 
ternary  alloys,  202. 
thermochemistry,  207. 
transference,  104. 
transference-number,  105. 
transition-temperature,  176,  189. 
triple-point,  177. 
uni  variant,  180. 
vapor-pressure,  47. 
variance,  180. 
velocity  of  molecules, 

mean,  34. 

mean  square,  34. 

most  probable,  34. 
viscosity,  118. 
voltaic  action,  98. 
voltaic  cell,  244. 
volume 

constant  b,  28. 

molal,  21. 

specific,  21. 
weight-normality,  55. 
work,  36. 
work-content,  227. 


302 


INDEX 


CONDUCTANCE,  electrical, 

actual,    specific,    and    equivalent, 

112. 

determination,  113. 
equivalent,  at  zero  concentration, 

114-116. 

general  discussion,  112. 
ion-,    determination    and    values, 

1 20. 
ion-concentration,    determination, 

121. 

mixtures,  116. 
relation  to 

activity,  171-172. 
concentration,  114-119. 
ion-mobility,     113-114,     116- 

119. 
ionization,  117-119,  152-154, 

171-172. 

transference,  113-114. 
viscosity,  118-119. 
Conduction, 
electrolytic 

Faraday's  law,  99-100. 
general  discussion,  98,  112. 
mechanism,  100-102,  106-107, 

113-119- 
metallic,  98,101. 
Conservation  of  energy,  law,  35,  207. 

CONSTANTS  OF  CHEMICAL  SUBSTANCES, 
activity-coefficients,  171. 
atomic  weights,  8. 
boiling-point     constants    for    sol- 
vents, 62. 

composition  of  air,  15. 
conductance-ratios  for  largely  ion- 
ized substances,  124,  171. 
density  of 

air,  14. 

mercury,  13. 

deviation-coefficients  for  gases,  19. 
electrode-potentials,  molal,  260. 
freezing-point  constant  for 

benzene,  83. 

water,  82. 
heat-capacities  of 

elementary  substances,  92. 

elements,  93. 

gases,  42. 

solid  compounds,  93. 

water- vapor,  216. 
heats  of  vaporization,  217. 
hydration  of  ions,  in. 
hydrogen-ion  concentration,  change 
during  neutralization,  163. 
indicator-constants,  164. 
ion-conductances,  120. 


CONSTANTS  OF  CHEMICAL  SUBSTANCES 

(continued) 
ionization-constants  of 

acids  and  bases,  123,  152,  157. 

poly  basic  acids,  157. 

water,  154. 

molal  weight  of  air,  14. 
molecular 

collisions,  30. 

diameters,  30. 

mean  free-path,  30. 
mol-numbers  of  salts,  96. 
overvoltages,  275. 
phase-diagrams  of  substances, 

acetic-acid  benzene,  191. 

alloys,  196,  199,  201. 

organic  liquids,  73. 

salt-hydrates,    188-189,    194- 

195- 

sulfur,  176. 
pressure-volume  product  of  gases 

at  moderate  pressure,  19. 

at  high  pressure,  19,  20,  28. 
solubilities  of  air-gases,  65. 
transference- numbers,  109. 
van  der  Waals'  constants,  28. 
vapor-pressure  of 

ice,  80. 

water,  48,  80. 
viscosities  of  salt  solutions,  119. 

CONTACT  agents,  141. 

Contact-process  for  sulfuric  acid,  15, 222. 

Cooling-curves,  192,  194. 

D ALTON'S  partial-pressure  law,  15. 
Deacon  chlorine  process,  221. 
Decomposition-potential,  272. 
Definite  proportions,  law,  4. 
Dehydration.     See  Drying. 
Diagrams.     See  Phase  diagrams. 
Diffusion-potentials. 

See  Liquid-potentials. 
Dilution,  heat  of,  218. 
Displacement  of  acids  or  bases, 

determination,  160. 

principles,  159-160. 
Dissociation  of  gases, 

determination    from   density,    16, 

148-149- 

mass-action  relations,  148-149. 
pressure,  effect  of,  149. 
Dissociation  of  solids, 

dissociation-pressure,  165,  187,  203. 
gas-phase    with    one    component, 

165-166,  187-190. 
gas-phase   with   two  components, 

167. 


INDEX 


303 


Dissociation  of  solids  (continued) 

gaseous  and  liquid  phases,  203. 

heat  of,  282. 

pressure-temperature  relations, 

165-166,  187-190. 
Distillation, 

of  solutions,  68,  69,  72-74,  79. 

steam,  48,  79. 
Distribution  of 

base  between  acids,  159-160. 

kinetic  energies,  31-34. 

molecular  velocities,  33,  34,  52,  53. 

solutes    between    phases,    63-66, 

76-77,  235. 
Dry  cell,  261. 
Drying  of 

organic  liquids,  190. 

salt  hydrates,  190. 
Dulong  and  Petit's  law,  91. 

EFFLORESCENCE  of  salt-hydrates,  190. 
Electro-analysis,  272. 
Electrochemical  equations,  102-103. 
Electrode-potentials, 

concentration  effect,  257-259. 

electromotive    force,    relation    to, 
260-261. 

equilibrium-constants,  relation  to, 
269-270. 

expression,  255-256. 

mechanism,  254. 

molal,  257-260. 

normal  calomel  cell,  259. 

temperature  effect,  291. 
Electrode, 

molal  hydrogen,  255. 

normal  calomel,  259. 

processes,  98-102,  254-255. 
Electrolysis, 

deposition  of  metals,  272. 

energy-efficiency,  275. 

Faraday's  law,  99-100. 

formulation,  102-103. 

mechanism,  100-102. 

polarization,  273-276. 

products,  98-99. 

separation  of  elements,  272. 
Electrometric  titration,  266-267. 
Electromotive  force, 

back,  determination,  274. 

cells  involving 

chemical  changes,  252-253. 
concentrated    solutions,    261- 

262. 
physical  changes,  249-251. 

concentration-cells,  250-251. 

relation  to 

activity,  250. 


Electromotive  force  (continued) 
relation  to 

concentration,  249,  271. 
electrode-potentials,  254,  260- 

261. 

equilibrium-constant,  252-253. 
free  energy,  247-253,  294-295. 
temperature,  290-291. 
Elements  and  elementary  substances, 
general  relations,  4. 
heat-capacity  of  gaseous,  41. 
heat-capacity  of  solid,  91-92. 
molecular  formulas  of  gaseous,  19, 

.    41-44. 
periodic  law,  95. 
separation  by  electrolysis,  272. 
Energy, 

conservation,  35. 

content  in  general,  39-40,  207-208, 

209-211. 
content  of 

imperfect  gases,  44-46. 
^perfect  gases,  40-44. 
efficiency  of  electrolysis,  275. 
general  discussion,  35-40. 
intramolecular,  43. 
molecular,  42-43. 
units,  37. 
Enzymes,  143. 
Equations, 

electrochemical,  102-103. 
free-energy,  292-293. 
heat-content,  211-214. 
of  state  of  gases,  13,  19,  21,  27. 
thermochemical,  211-214. 
Equilibrium  of  gas  or  solute  reactions, 
derivation  from  reaction-rate,  138. 
derivation  from  free-energy,  238- 

242. 

determined  thermochemically,  221. 
general  discussion,  145,  174-175. 
oxidation-reactions,  269-270. 
relation  to 

electromotive  force,  252-253. 
free-energy,     236,    240,    242, 

295-296. 
pressure,  150. 
temperature,  285-289. 
(See  also  Mass-action  law.andPhase 

equilibria.) 
Equilibrium  of  polyphase  systems, 

general    considerations,     174-175, 

202-203. 

phase  rule,  180-186. 
relation  to 

activity,  236. 

pressure     and     temperature, 
280-283. 


304 


INDEX 


Equilibrium     of     polyphase     systems 

(continued) 
systems  with 

one-component,  176-179. 
solid  solutions,  197-200. 
three  components,  201-203. 
two  components,  187-200. 
two  liquids,  76-79. 
unstable  conditions,  177-179,  195. 
(See    also    Phase    diagrams,    and 

Phase  rule.) 

Equilibrium-constants,  relation  to 
dissociation-constants,  150. 
electrode-potentials,  269-270. 
electromotive  force,  252-253,  266- 

267,  269-270. 

free-energies,  240,  242,  295-296 
ionization-constants,  157. 
oxidation-reactions,  269-270. 
reaction-rate,  138. 
Eutectic 

composition,  192-193,  202. 
point,  192,  202. 
texture,  192. 
Explosions,  pressure  and  temperature, 

221. 
Extraction  by  solvents,  77. 

FARADAY'S  law,  99-100. 

Flames,  maximum  temperature,  221. 

Formation, 

free  energy  of,  213-215. 
heat  of,  213-215. 
Free  energy, 

activity,  relation  to,  237. 
change  attending 

activity  changes,  234. 
compression  of  liquids,  231. 
concentration-changes,      231- 

expansion  of  gases,  230. 
formation  of  compounds,  294- 

295- 
reactions  between  gases,  238- 

240. 
reactions  between  solutes,  240- 

241. 

reactions  involving  solids,  242. 
transfer  between  phases,  234- 

237. 
transfer     between     solutions, 

231-234. 
transition  of  unstable  phases, 

235-236. 

voltaic  action,  247-253. 
volume  changes,  230-231. 
concept,  226-229. 
electromotive  force,  relation,  247- 

253.  294-295. 


Free  energy  (continued) 

elementary  substances,  292,  294. 

equations,  231,  234,  239,  241,  242, 
284-285,  293. 

equilibrium,  relation  to,  236,  240, 
242,  295-296. 

expression  by  equations,  293. 

importance,  228-229,  292. 

ions,  293. 

of  formation,  292-293. 

temperature,  effect  of,  284-285, 295. 

values,  determination,  294-296. 

values,  systemization,  292-296. 
Freezing-point, 

constants,  81-82. 

heat  of  fusion,  relation,  81,  82. 

lowering,     composition     relations, 
81-83. 

lowering,  effect  of  salts,  96-97. 

lowering  of  metals,  193. 

pressure,  effect  of,  177,  281. 

raising  by  solutes,  198. 

solid  solutions,  198-200. 

vapor-pressure  relations,  80-82. 
Fusion,  heat  of, 

freezing-point  relation,  81,  82. 

heat-of-vaporization  relation,  81. 

GAS-CONSTANT,  calculation  of,  13,  38. 
Gases, 

cohesion,  27,  28,  46. 
dissociation,  16,  148-149. 
energy  of  imperfect,  44-46. 
energy  of  perfect,  40-44. 
kinetic  theory,  22-34,  42-46. 
mass-action     relations,     148-151, 

165-166. 

partial  pressure,  14-16. 
reactions, 

equilibrium,  temperature-effect, 

285-288. 
heat  of,  209. 
rate  of,  135-136. 
solubility,  64-66. 
volume 

at  high  pressures,  19,  20. 
at  moderate  pressures,  19, 25-28. 
of  perfect,  10-16,  22-25. 
Gibbs-Helmholtz  equation,  290. 

HEAT-CAPACITY, 

additive  property,  93. 
general  discussion,  39. 
influence  on  heat  of  reaction,  216. 
of  perfect  gases, 

kinetic  interpretation,  42-44. 

physical  laws,  40,  41. 

relation  to  atomic  weight,  42. 

relation  to  composition,  41,  42. 


INDEX 


305 


HEAT-CAPACITY  (continued) 
of  solid  substances 
compound,  93. 
elementary,  91-92. 
ratio  for  gases,  41. 
Heat-content  decrease, 
attending 

aqueous-solution  reactions, 

218-220. 

combustions,  209. 
dilution,  218. 
formation  of  compounds,  213- 

216. 

gaseous  reactions,  209. 
ionization,  220.  , 

neutralization,  219-2267- 
solution,  217-218. 
transition,  218. 

determination,  208,  214-215,  290. 
energy-content  relation,  210. 
expression  by  equations,  211-214, 

286. 

temperature-effect,  215-216,  286. 
Heat  of  fusion.     See  Fusion. 
Heat  of  reaction  of  various  types. 

See  Heat-content  decrease. 
Heat  of  vaporization.    See  Vaporization. 
Henry's  law,  63-66. 
Hydrates  of  salts.    See  Salt-hydrates. 
Hydration  of 

ions,  in,  121. 
salts,  heat  of,  281. 
solutes,  65. 
Hydrogen-ion, 

acid  salts,  142,  158. 
catalytic  effect,  142. 
change  on  neutralization,  163. 
concentration  in  water,  122. 
determination,  142,  158,  159,  162. 
Hydrolysis  of  salts, 
constant,  156-157. 
determination,  122,  155,  267. 
mass-action  relations,  156-157. 
temperature-effect,  289. 
Hygroscopicity  of  salts,  190. 

INDICATORS, 

constants  of,  determination,   162, 
267. 

constants,  values,  164. 

principles,  161-164. 
Ionic  theory,  explanation  of 

conductance,  113-119. 

electrode-potentials,  254. 

electrolysis,  101-103. 

electromotive  force,  254-255. 

heat  of  reaction,  220. 


Ionic  theory,  explanation  of  (continued) 

largely    ionized    substances,    123- 
126,  171. 

liquid-potentials,  262. 

mass-action  effects,  152-164,  167- 
173- 

molal  properties  of  salts,  96-97. 

transference,  106-108. 

voltaic  action,  243-244. 
Ions  or  ion-constituents, 

activity,  171-172. 

complex,  78,  in,  167,  169,  267. 

composition  determined,  in. 

concentration,    determined,    121- 
122,  266-267. 

concentration  in  water,  122. 

conductance,  120-121. 

electric  charge  on,  101. 

electron  relations,  101. 

electrostatic  separation,  255. 

free  energy,  293. 

general  discussion,  100. 

hydra tion,  in,  121. 

ionization,  106,  117-119,  123-126. 

mobility.     See  Mobility,  ionic. 

transference,  107. 
Ionization, 

acids  and  bases,  123,  152. 

acid  salts,  157-158. 

common-ion  effect,  152. 

dibasic  acids,  157-159. 

from  conductance,  117-119,  152. 

from  molal  properties,  97. 

general  discussion,  96,   106,   123- 
.    126,  153. 

heat  of,  220. 

hypotheses,  124-126,  153. 

mass-action  relations,  152-154. 

salts,  123-126,  153. 

water,  122,  154,  157. 
lonization-constants, 

acids  and  bases,  152. 

determination,  152,  154,  157,  159, 
267. 

dibasic  acids,  157. 
Irreversible  processes,  224-225. 

JouLE-Thomson  effect,  44. 

KINETIC 

derivation  of  Clapeyron  equation, 

53-54- 

energy  of  molecules,  24,31  -34, 4  2 » 43 • 
hypotheses,  22. 

interpretation  of  vaporization,  si- 
interpretation  of  vapor-pressure  of 

solutions,  58,  70,  71. 


306 


INDEX 


KINETIC  (continued) 

mechanism  of  reaction-rate,  135, 
theory  of  imperfect  gases,  25-28, 

45-46. 
theory  of  perfect  gases,  22-25,  42- 

44- 
Kopp's  law  of  heat-capacity,  93. 

LAWS,   PRINCIPLES,   FUNDAMENTAL 

EQUATIONS, 
Avogadro's,  12,  24. 
boiling-point  raising,  60,  61. 
Boyle's,  10. 
Clapeyron  equation, 

approximate,  50. 

exact,  49,  280. 
combining  volumes,  12. 
conservation  of  energy,  35,  207. 
constant  energy-content  of  gases, 

40. 

Dalton's  partial  pressure,  15. 
definite  proportions,  4. 
distribution  between  phases,  77. 
Dulong  and  Petit's,  91. 
Faraday's,  100. 

free-energy    equations,    231,    234, 
239,  241,  242,  284, 285, 293. 
freezing-point  lowering,  81,  82. 
gases,  at  moderate  pressure,  19. 
gases,  perfect,  10-13. 
Gibbs-Helmholtz  equation,  290. 
heat-content  equation,  212. 
Henry's,  63. 
indicator  equation,  162. 
initial  and  final  states,  36. 
independence  of  reaction-rates,  137. 
kinetic  equation  for 

distribution  of  energies,  33. 

imperfect  gases,  27. 

perfect  gases,  22. 

vapor-pressure,  54. 
Kopp's  heat-capacity,  93. 
liquid-potential  equations,  263. 
mass-action    law    of    equilibrium, 

146,  148. 
mass-action  law  of  reaction-rate, 

131.  134- 

Maxwell's  distribution,  32. 
Ohm's,  112. 
osmotic  pressure,  87. 
perfect  solutions,  56. 
periodic,  95. 

perpetual  motion,  35,  76,  223. 
phase  rule,  181. 
Raoult's,  57,  58,  59. 
salting-out,  66. 
solubility  of  gases,  64. 
solubility-lowering,  78. 
solubility-product,  167. 


LAWS,    PRINCIPLES,    FUNDAMENTAL 
EQUATIONS  (continued) 

thermodynamics,  first  law,  35. 

thermodynamics,  second  law,  224. 

transference,  105. 

Trouton's  rule,  217. 

van  der  Waals  equation,  21,  27. 

van't  Hoff  equation,  286. 
Lime-burning,  187. 
Liquid  air  processes,  45. 
Liquid-potentials, 

conventions,  256. 

derivation  of  equations,  264-266. 

equations  for,  263. 

mechanism,  262. 
Liquids, 

boiling-point,  48. 

compression  of,  231. 

heat  of  vaporization,  49,52-54,  217. 

mixtures  of.     See  Solutions. 

partially  miscible,  79. 

pressure,  effect  on  vapor-pressure, 
85-86. 

temperature-change  on  mixing,  70. 

vapor-pressure,  47-54,  85-86. 

volume-change  on  mixing,  70. 

MASS-ACTION  law  of  equilibrium, 
applied  to 

equilibrium  in  general,    146- 

147,  165. 

imperfect  solutes,  169-173. 
indicators,  161-164. 
perfect  gases,  148-151. 
perfect  solutes,  152-169. 
solids  and  gases,  165-166. 
solids  and  solutes,  166-169. 
titration,  162-164. 
derivation,  138,  240,  241. 
formulation,  146,  148,  167. 
phase-rule,  contrasted,  203. 
Mass-action  law  of  reaction-rate,  131- 

138.    ^ 

Maxwell's  distribution  law,  32. 
Melting-point.     See  Freezing-point. 
Metals, 

deposition  by  electrolysis,  272. 
polarization,  275. 
(See  also  Elements,  and  Alloys.) 
Molal  properties  of 
gases,  10-46. 

ionized  substances,  96-97. 
solutions,  57-90. 
Molal  quantities,  21. 
Mobility  (ionic) 

conductance,  relation  to,  113. 
experimental  determination  of,  107- 
109. 


INDEX 


307 


Mobility  (ionic)  (continued) 

general  discussion,  106-107,  123- 
126. 

transference,  relation  to,  107-108. 

viscosity,  effect  of,  118-119. 
Molecular 

formulas,  18. 

formulas  of  elementary  substances, 
19. 

magnitudes,  30-31. 

mechanism  of  electromotive  force, 

243- 
mechanism  of  reactions,  134-1351 

140-142. 
nature  of  gases,  12,  18,  19,  22-34, 

41-44. 

nature  of  solutes,  57,  63,  65,  70-72. 
species.     See  Chemical  substance, 
weight 

effect  of  solvent,  63,  83. 
of  gases,  12,  16,  17. 
of  solutes,  57,  62,  63,  83. 
(See  also  Molecules  and  Kinetic.) 
Molecules, 

atomic  composition,  19,  41-44- 
attraction,  26-28,  46,  51,  70,  71. 
collisions    with    one   another,  30, 

3i- 

diameter,  29-30. 

distribution  of  energies,  31-34- 

distribution  of  velocities,  33-34- 

free-path,  30. 

impacts  on  unit-surface,  23,  26,  53. 

kinetic  energy,  24,  31-34*  42- 

mass,  25. 

number  in  one  mol,  25. 

rotational  energy,  43. 

velocity,  24,  30,  33,  34,  52. 

volume,  26-29. 
Mol-number,  96. 

Moving-boundary  method  of  transfer- 
ence, 108. 

NEUTRALIZATION, 

heats  of,  219-220. 

indicators,  161-164. 
Nitrogen-fixation,  arc  process,  136,  287. 

OHM'S  law,  112. 
Order  of  reactions,  133. 
Osmotic  pressure, 

experimental  determination,  84. 
general  discussion,  84. 
relation  to 

composition,  87-88. 
hydrostatic  pressure,  88. 
vapor-pressure,  85-87. 
Overvoltage.    See  Polarization. 


Oxidation-reactions 

equilibrium  in  relation  to  electrode- 
potentials,  269-270. 
potentials  of,  270. 

PERIODIC  law,  95. 
Perpetual  motion, 
first  kind,  35. 
phase-equilibrium  derivations,  76- 

78. 

phase-rule  derivation,  183-186. 
second  kind,  76,  223. 
Phase  diagrams, 

boiling-point  composition,  68,  73. 
pressure-temperature,     176,     188- 

189,  196. 
substances 

acetic-acid  benzene,  191. 
alloys,  193,  ^194,  196,  199-202. 
organic  liquids,  73. 
salt-hydrates,    188-189,    194- 

195- 
sulfur,  176. 

temperature-composition,    68,    73, 
190-197,  199-202. 

triangular,  201. 

vapor-pressure  composition,  68,  72, 
79,  188. 

vapor-pressure  freezing-point,  80. 
Phase  rule, 

component-concept,  174,  181-183. 

derivation  deductively,  183-186. 

derivation  inductively,  180-181. 

mass-action  law,  contrasted,  203. 

statement,  181. 

variance-concept,  180. 
Phase  equilibria. 

See  Equilibrium  of  polyphase  sys- 
tems. 
Polarization, 

concentration-,  271-273. 

gas-,  273-276. 

Porous-plug  experiments,  44. 
Pressure  of  gases, 

cohesion,  27,  28,  46. 

height,  change  with,  86. 

kinetic  interpretation,  22-29. 

partial,  14-16. 

volume  and. temperature  relations, 
10-21. 

wet  and  dry,  47. 

Probability  of  kinetic  energies,  3I~34- 
Producer-gas  process,  166. 

RAOULT'S  law.  57-59- 

Rate  of  reactions, 

catalytic  effects,  140-143- 
concentration  effects,  131-139- 
dissolving  of  solid  substances,  139- 


308 


INDEX 


Rate  of  reactions  (continued) 

mass-action  law,  131-138. 

mechanism,  134-135,  140-142. 

order  of  reactions,  132-134. 

simultaneous  reactions,  136-138. 

surface  effects,  139. 

temperature  effects,  144. 

transition  of  solids,  178-179. 
Reduction-potentials,  270. 
Reversible  processes,  224-226. 

SALT-hydrates, 

determination  of  existing,  189,  195. 

diagrams,  188-189,  194-195. 

drying,  190. 

efBorescenceandhygroscopicity,i9O. 

heat  of  hydration,  281. 

preparation,  195. 

solubility,  194-196. 

transition,  189,  190. 

unstable  forms,  195-196. 

vapor-pressures,  188-189,  196. 
Salting-out  effect,  66,  77. 
Salts.    See  Substances,  largely  ionized. 
Saturation  of  solution,  rate,  139. 
Semipermeable  walls,  14,  84. 
Shaking-out  by  solvents,  77. 
Solid  solutions, 

description,  197. 

freezing-point,  1 98-20x5. 
Solid  substances, 

atomic  properties,  91. 

heat-capacity,  91-94. 

rate  of  dissolving,  139. 

solubility,  166-169,  I73- 
Solubility  of 

gases,  64-66.    . 

largely  ionized  substances,  167-169, 

173- 
salts,  determined  from 

conductance,  121. 

electromotive  force,  267. 
salts,  effect  of 

common-ion,  168. 

complex  formation,  167,  169. 

metathesis,  168-169. 

temperature,  289. 
solvents  in  one  another,  78,  79. 
unstable  phases,  177,  179. 
Solution,  heat  of,  217-218,  220,  291. 
Solutions  (and  solutes), 

boiling-point  of  concentrated,  67- 

69,  73-75- 

boiling-point  of  dilute,  59-62. 
classes,  56,  67. 

composition,  expression  of,  55,  56. 
conductance,  112-122. 
electrolysis,  98-103. 


Solutions  (and  solutes)  (continued) 
freezing-point,  80-83. 
ionization,  96,  123-126. 
molal  properties,  57-90. 
osmotic  pressure,  84-88. 
Raoult's  law  deviations,  70,  71. 
reactions, 

equilibrium  of. 

See  Equilibrium, 
rate  of,  131-135,  139,  144- 
solid,  197-200. 

thermochemistry  of,  217-221. 
transference,  104-111. 
vapor-pressure     of     concentrated, 

67-72,  79.^ 

vapor-pressure  of  dilute,  57-66. 
Specific  heat.     See  Heat-capacity. 
Standard  cell,  Clark,  262. 
Steam  distillation,  48,  79. 
Storage  cells,  245-246,  261. 
Substances, 

chemical,  classes  of,  3,  4,  6. 
largely  ionized, 

activity,  171-172. 
concentration  determined  from 

conductance,  121. 
conductance,  115-116,  119. 
heat  of  reactions,  219-220. 
ionization,    96-97,    119,    123- 

126,  153,  171-173. 
solubility,  167-169. 
slightly  ionized, 

conductance  at  zero  concentra- 
tion, 115. 

ionization,  118,  123. 
(See  Acids  and  bases.) 

TEMPERATURE-change  produced  by 

expansion  of  gases,  45. 

flames  and  explosions,  221. 

mixing  of  liquids,  70. 
Temperature-composition  diagrams. 

See  Phase  diagrams. 
Tempering,  197. 
Thermochemical 

applications,  221-222. 

equations,  211-214. 

principles,  207-216. 

results,  217-221. 
Thermodynamics, 

first-law,  35-37,  207-208. 

second-law,  76,  223-229,  277-279. 
Titration  of  acids  and  bases, 

electrometric,  267-268. 

polybasic  acids,  164. 

principles,  162-164. 
Transference  (electrical), 

concentration,  effect  of,  109-110. 


INDEX 


309 


Transference  (electrical)  (continued) 

conductance,  relation  to,  113-114. 

determination   of    composition    of 
ions,  iio-ui. 

experimental  determination,   104- 
106,  108. 

general  discussion,  104. 

ion-mobility,  relation  to,  106-108. 

mechanism,  106-107. 

moving-boundary  method,  108-109. 

numbers,  105-106. 
Transition 

catalysis  of,  178-179. 

free  energy  of,  234-236. 

heat  of,  218,  281. 

salt-hydrates,  189. 

temperature,  determination,  178. 

temperature,  effect  of  pressure,  177, 
281. 

unstable  phases,  177-178. 
Trouton's  rule,  217. 

UNITS,  definitions, 

ampere,  100. 

bar,  13. 

calorie,  37. 

coulomb,  100. 

dyne,  13. 

erg,  37- 

faraday,  100. 

joule,  37. 

megabar,  13. 

mho,  112. 

ohm,  112. 

mol,  12. 

reciprocal  ohm,  112. 
Univariant  systems,  temperature-pres- 
sure relations,  280. 
Unstable  phases, 

diagrams,  176-178. 

discussion,  177-179. 

free-energy,  235-236. 

solubility,  177,  179,  195-196- 

transition,  178-179. 

vapor-pressure,  177. 

VAN  DER  WAALS  equation,  21,  27. 
Van't  Hoff  equation,  285-289. 
Vapor-density,  determination,  16. 
Vaporization,  heat  of, 

boiling-point  relations,  49,  60,  61, 
217. 

kinetic  interpretation,  52-54. 

Trouton's  rule,  217. 

vapor-pressure  relation,  48-50,  282. 
Vapor-pressure 

composition  diagrams,  68,  72,  79, 
189. 

determination,  47,  67. 


law), 


Vapor-pressure  (continued) 
general  discussion,  47. 
molecular  interpretation,  51-54, 

58,  70,  71. 
of  various  systems, 

partially  miscible  liquids,  79. 
perfect  solutes,  63-66. 
salt-hydrates,  188-189,  196. 
solutions,  concentrated,  67-72. 
solutions,  dilute,  57-59,  63-66. 
solvents,  57-59. 
unstable  phases,  177. 
relation  to 

activity,  170. 
boiling-point,  48,  59-61. 
composition    (Raoult's 

57-59,  67-72,  79- 
freezing-point,  80-82. 
heat  of  vaporization,  49,  54. 
osmotic  pressure,  85,  87. 
pressure,  85-86. 
temperature,  48-50. 
Variance, 

applications,  180-181^ '203. 
concept,  1 80. 
Viscosity, 

conductance,  effedt  on,  118-119. 
general  discussiop,  118. 
ion-mobility,  effect  on,  118-119. 
Voltaic  cells, 

changes  in  state,  243-247,  252. 
concentration-cells,  250-251. 
conduction  in,  99-102. 
electrode-potentials,  254-261. 
electrode  products,  98-99. 
electromotive  force,  249-268. 
formulation,  102-103,  244-247. 
free  energy  decrease,  247-253. 
involving 

chemical  changes,  252-253. 
concentrated    solutions,    261- 

262. 

physical  changes,  240-251. 
liquid-potentials,  262-266. 
mechanism  of  action,  243-244. 
polarization,  273-276. 
standard  cell  (Clark),  262. 
storage  cells,  245-246,  261. 
work  produced,  247-248. 
Volume-change  attending 

compression  of  liquids,  231. 
displacement  of  acids,  160. 
mixing  of  liquids,  70. 
neutralization,  160. 

WATER 

as  catalyst,  143. 
gas  process,  150-151. 
ionization,  122,  154,  267,  289. 


3io 


INDEX 


Work-content, 

effect  of  temperature,  284-285. 

general  discussion,  226-228. 
Work,  produced  by 

changes  in  general,  223-229. 

gas  reactions,  209,  211. 

heat-transfer     between     tempera- 
tures, 277-279. 


Work,  produced  by  (continued) 
reversible  changes,  226-228. 
voltaic  cells,  247-253. 
volume-changes  in  general,  37,  38. 
volume-changes  of  perfect  gases, 
38. 


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